A164655, W. Lang, 20 Aug 2009 Partial sums for Theta(s):=sum(1/(2*k-1)^s,k=1..infty), for s=1,2,3,... Theta(s) = (1-1/2^s)*Zeta(s) (Zeta(s) is the Euler-, Riemann-Zeta function). Definition: Theta(s,N):= sum(1/(2*k-1)^s,k=1..N), N>=1. Theta(s,N) = Zeta(s,2*N) - Zeta(s,N)/2^s, with Zeta(s,N):= sum(1/k^s,k=1..N), N>=1. For Zeta(s,N), s=1,2,3,... see the W. Lang link under A103345. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s=1: Theta(1,n), n=1..40: [1, 4/3, 23/15, 176/105, 563/315, 6508/3465, 88069/45045, 91072/45045, 1593269/765765, 31037876/14549535, 31730711/14549535, 744355888/334639305, 3788707301/1673196525, 11552032628/5019589575, 340028535787/145568097675, 10686452707072/4512611027925, 10823198495797/4512611027925, 10952130239452/4512611027925, 409741429887649/166966608033225, 414022624965424/166966608033225, 17141894231615609/6845630929362225, 743947082888833412/294362129962575675, 750488463554668427/294362129962575675, 35567319917031991744/13835020108241056725, 250947670863258378883/96845140757687397075, 252846595191840484708/96845140757687397075, 13497714685925233086599/5132792460157432044975, 13591038185200822760144/5132792460157432044975, 13681087175729900515319/5132792460157432044975, 812316935828221562448796/302834755149288490653525, 49854167840670803800030081/18472920064106597929865025, 50147388794069321227488256/18472920064106597929865025, 50431587564286345811024641/18472920064106597929865025, 3397389286871291767268515972/1237685644295142061300956675, 3415326759977018463809109547/1237685644295142061300956675, 243725885602663452991747734512/87875680744955086352367923925, 17879865329739387154749952543301/6414924694381721303722858446525, 17965397658997810105466257322588/6414924694381721303722858446525, 1640791669937677803334857936583/583174972216520118520259858775, 130205716897293066581974036848832/46070822805105089363100528843225]. Numerators: s=1, n=1..40 [1, 4, 23, 176, 563, 6508, 88069, 91072, 1593269, 31037876, 31730711, 744355888, 3788707301, 11552032628, 340028535787, 10686452707072, 10823198495797, 10952130239452, 409741429887649, 414022624965424, 17141894231615609, 743947082888833412, 750488463554668427, 35567319917031991744, 250947670863258378883, 252846595191840484708, 13497714685925233086599, 13591038185200822760144, 13681087175729900515319, 812316935828221562448796, 49854167840670803800030081, 50147388794069321227488256, 50431587564286345811024641, 3397389286871291767268515972, 3415326759977018463809109547, 243725885602663452991747734512, 17879865329739387154749952543301, 17965397658997810105466257322588, 1640791669937677803334857936583, 130205716897293066581974036848832] Note that this is n o t A025550 (see the P. Barry/ T.D. Noe remark on A025550(39). Denominators: s=1, n=1..40 [1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675, 13835020108241056725, 96845140757687397075, 96845140757687397075, 5132792460157432044975, 5132792460157432044975, 5132792460157432044975, 302834755149288490653525, 18472920064106597929865025, 18472920064106597929865025, 18472920064106597929865025, 1237685644295142061300956675, 1237685644295142061300956675, 87875680744955086352367923925, 6414924694381721303722858446525, 6414924694381721303722858446525, 583174972216520118520259858775, 46070822805105089363100528843225] Note that this is n o t A025547 (see the P. Barry/ T.D. Noe remark on A025547(39). The rational sequence Theta(1;n) diverges. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s=2: Theta(2,n), n=1..40: A120268(n)/A128492(n). [1, 10/9, 259/225, 12916/11025, 117469/99225, 14312974/12006225, 2430898831/2029052025, 487983368/405810405, 141433003757/117279207045, 51174593563322/42337793743245, 51270597630767/42337793743245, 27164483940418988/22396692890176605, 3400039831130408821/2799586611272075625, 30634921277843705014/25196279501448680625, 25789165074168004597399/21190071060718340405625, 24804577707336170758506064/20363658289350325129805625, 24823277118070193095631689/20363658289350325129805625, 3548557216084587389770102/2909094041335760732829375, 4860883922861135897328099013/3982549742588656443243414375, 4863502298760273738513663388/3982549742588656443243414375, 8179529913958608810884711569603/6694666117291531481092179564375, 15130645477026759222806923871760322/12378437650872041708539440014529375, 15136758285743239243403733471767497/12378437650872041708539440014529375, 33449477490857687530387386679148930248/27343968770776340134163622992095389375, 11477077060617154871513468434232668130689/9378981288376284666018122686288718555625, 11480682974222835872922933302624167061314/9378981288376284666018122686288718555625, 32258617455880322251706537769757573993786651/26345558439048983626844906625785010422750625, 32267326731397363238029461705832213666653676/26345558439048983626844906625785010422750625, 32275435552092669173279337023353230905934301/26345558439048983626844906625785010422750625, 112377136715273630375812217084918381793980052406/91708888926329512005047119964357621281594925625, 418247034606459508140402306892945656276681369928351/341248775694872114170780333387374708788814718250625, 418333013133971443669293911915215088050122744258976/341248775694872114170780333387374708788814718250625, 418413782074964312808742617319567129401315363127201/341248775694872114170780333387374708788814718250625, 1878600716510209672312616389480924218591293479796255914/1531865754094280920512632916575925067752989270227055625, 1878922469451628340852946705153382930010691293127536539/1531865754094280920512632916575925067752989270227055625, 9473180034259752747160216973594779275251647797926138748724/7722135266389270120304182532459238266542818911214587405625, 50490298537836611659737100434819037996082573934059607979355821/41151258834588420471100988715475280722406681977862536284575625, 10099522863436996586897503677673691164753311468837801152716838/8230251766917684094220197743095056144481336395572507256915125, 10100910998327689399192025552820106252650845848395158634122963/8230251766917684094220197743095056144481336395572507256915125, 63048015792330027224451651672893378178938410276229757542818327208/51365001277333266432028254114656245397708020444768017790407295125]. Numerators: s=2, n=1..40, give A120268. [1, 10, 259, 12916, 117469, 14312974, 2430898831, 487983368, 141433003757, 51174593563322, 51270597630767, 27164483940418988, 3400039831130408821, 30634921277843705014, 25789165074168004597399, 24804577707336170758506064, 24823277118070193095631689, 3548557216084587389770102, 4860883922861135897328099013, 4863502298760273738513663388, 8179529913958608810884711569603, 15130645477026759222806923871760322, 15136758285743239243403733471767497, 33449477490857687530387386679148930248, 11477077060617154871513468434232668130689, 11480682974222835872922933302624167061314, 32258617455880322251706537769757573993786651, 32267326731397363238029461705832213666653676, 32275435552092669173279337023353230905934301, 112377136715273630375812217084918381793980052406, 418247034606459508140402306892945656276681369928351, 418333013133971443669293911915215088050122744258976, 418413782074964312808742617319567129401315363127201, 1878600716510209672312616389480924218591293479796255914, 1878922469451628340852946705153382930010691293127536539, 9473180034259752747160216973594779275251647797926138748724, 50490298537836611659737100434819037996082573934059607979355821, 10099522863436996586897503677673691164753311468837801152716838, 10100910998327689399192025552820106252650845848395158634122963, 63048015792330027224451651672893378178938410276229757542818327208] Denominators: s=2, n=1..4, give A128492. [1, 9, 225, 11025, 99225, 12006225, 2029052025, 405810405, 117279207045, 42337793743245, 42337793743245, 22396692890176605, 2799586611272075625, 25196279501448680625, 21190071060718340405625, 20363658289350325129805625, 20363658289350325129805625, 2909094041335760732829375, 3982549742588656443243414375, 3982549742588656443243414375, 6694666117291531481092179564375, 12378437650872041708539440014529375, 12378437650872041708539440014529375, 27343968770776340134163622992095389375, 9378981288376284666018122686288718555625, 9378981288376284666018122686288718555625, 26345558439048983626844906625785010422750625, 26345558439048983626844906625785010422750625, 26345558439048983626844906625785010422750625, 91708888926329512005047119964357621281594925625, 341248775694872114170780333387374708788814718250625, 341248775694872114170780333387374708788814718250625, 341248775694872114170780333387374708788814718250625, 1531865754094280920512632916575925067752989270227055625, 1531865754094280920512632916575925067752989270227055625, 7722135266389270120304182532459238266542818911214587405625, 41151258834588420471100988715475280722406681977862536284575625, 8230251766917684094220197743095056144481336395572507256915125, 8230251766917684094220197743095056144481336395572507256915125, 51365001277333266432028254114656245397708020444768017790407295125] The limit of the sequence Theta(2,n) is (Pi^2)/8 = (1-1/2^2)*Zeta(2) approximately 1.233700550 (Maple12, 10 digits). The values Theta(2,10^k), k=0..4 are: [1., 1.208721311, 1.231200571, 1.233450550, 1.233675550] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s=3: Theta(3,n), n=1..40: [1, 28/27, 3527/3375, 1213136/1157625, 32797547/31255875, 43684790932/41601569625, 96017087247229/91398648466125, 96044168328256/91398648466125, 471956397645187853/449041559914072125, 3237597973008257555852/3079976059450620705375, 462561506842656976961/439996579921517243625, 5628425850334528955928112/5353438387905100303185375, 703596058798919360293439483/669179798488137537898171875, 18998011529681231695738912916/18067854559179713523250640625, 463360571051954739540899597748949/440656904843834033118559874203125, 13804415429113627479696058476413142784/13127609852202659680635017212385296875, 13804780724209272710507774396417939659/13127609852202659680635017212385296875, 96635608352706084917992075471613313988/91893268965418617764445120486697078125, 4894975363158586737968817043984115890512289/4654669752905349245622438688012667098265625, 4895053831604664813277666189092563650371664/4654669752905349245622438688012667098265625, 337376659797778008945155653857136592014363720169/320804494039989575357544096816521029079565140625, 26824126895035976146777848115316175542315095864617308/25506202907637451167952258505591137459028985635671875, 26824406798497239657901776260814179285936045428245683/25506202907637451167952258505591137459028985635671875, 2785015893243286550453504068985016127141197073482387217984/2648130504479643092610307334825988664408766375652361078125, 955268171879544895094451634167209878515157735465029230815387/908308763036517580765335415845314111892206866848759849796875, 955275019236617487970349782959272096772931082118027314206012/908308763036517580765335415845314111892206866848759849796875, 142219387347652938274142529973043397265375552919352401216898245399/135226283714587627871600840704802829036176081715842820158209359375, 142220200127720343759509151165173854382304395450271805230843599024/135226283714587627871600840704802829036176081715842820158209359375, 142220930318838337353369996882024409224166864204931624137714583399/135226283714587627871600840704802829036176081715842820158209359375, 29209327674236413474925648190473995944879202579626367876599841633262596/27772638923018292424641509063111700224620807486718082561272880019078125, 6629990177464778385244523193431041185264850901533659325081069926639596380801/6303861355385615032837554368654156828684655504142757097840279579610371890625, 6630015388170587235640780047359977161928552864186697125003075950625405490176/6303861355385615032837554368654156828684655504142757097840279579610371890625, 6630038342604289140474763980193363248981183486509880166449039866072213356801/6303861355385615032837554368654156828684655504142757097840279579610371890625, 1994076525898049200371644276529265165010156373608675231258810417519056716203429788/1895968252829843734121321364579530170265683043392488053017736007202354280940046875, 1994082297332782586195880815928077920612499888346607502484417684705061368600414163/1895968252829843734121321364579530170265683043392488053017736007202354280940046875, 713705885088926378051688020030998876174510713221066030309753435686480421851423773540368/678587893338582200723096250918024222769960883743648791543630912073801823045533117078125, 277644400887532211392734241584650007837002404086003187561800895921359628069198367644470416381/263982226502895231978696734243377029069301873111303021939928666522214163795704156606380953125, 277645026623180218255506339236168214360330568357109821761556605381902495539808846350619409284/263982226502895231978696734243377029069301873111303021939928666522214163795704156606380953125, 277645604855699012943364563497578108884995968890856804571171550914134606283531196551276549909/263982226502895231978696734243377029069301873111303021939928666522214163795704156606380953125, 136890375354675488537815499719016656603578596807852259371987790219820534361589733320601445271536576/130153532972760962279544659154620367035299526216923730634240489813445949103670181669053458747796875] Numerator Theta(3,n), n=1..40: A164655 [1, 28, 3527, 1213136, 32797547, 43684790932, 96017087247229, 96044168328256, 471956397645187853, 3237597973008257555852, 462561506842656976961, 5628425850334528955928112, 703596058798919360293439483, 18998011529681231695738912916, 463360571051954739540899597748949, 13804415429113627479696058476413142784, 13804780724209272710507774396417939659, 96635608352706084917992075471613313988, 4894975363158586737968817043984115890512289, 4895053831604664813277666189092563650371664, 337376659797778008945155653857136592014363720169, 26824126895035976146777848115316175542315095864617308, 26824406798497239657901776260814179285936045428245683, 2785015893243286550453504068985016127141197073482387217984, 955268171879544895094451634167209878515157735465029230815387, 955275019236617487970349782959272096772931082118027314206012, 142219387347652938274142529973043397265375552919352401216898245399, 142220200127720343759509151165173854382304395450271805230843599024, 142220930318838337353369996882024409224166864204931624137714583399, 29209327674236413474925648190473995944879202579626367876599841633262596, 6629990177464778385244523193431041185264850901533659325081069926639596380801, 6630015388170587235640780047359977161928552864186697125003075950625405490176, 6630038342604289140474763980193363248981183486509880166449039866072213356801, 1994076525898049200371644276529265165010156373608675231258810417519056716203429788, 1994082297332782586195880815928077920612499888346607502484417684705061368600414163, 713705885088926378051688020030998876174510713221066030309753435686480421851423773540368, 277644400887532211392734241584650007837002404086003187561800895921359628069198367644470416381, 277645026623180218255506339236168214360330568357109821761556605381902495539808846350619409284, 277645604855699012943364563497578108884995968890856804571171550914134606283531196551276549909, 136890375354675488537815499719016656603578596807852259371987790219820534361589733320601445271536576]. Denominator Theta(3,n), n=1..40: looks like A128507. [1, 27, 3375, 1157625, 31255875, 41601569625, 91398648466125, 91398648466125, 449041559914072125, 3079976059450620705375, 439996579921517243625, 5353438387905100303185375, 669179798488137537898171875, 18067854559179713523250640625, 440656904843834033118559874203125, 13127609852202659680635017212385296875, 13127609852202659680635017212385296875, 91893268965418617764445120486697078125, 4654669752905349245622438688012667098265625, 4654669752905349245622438688012667098265625, 320804494039989575357544096816521029079565140625, 25506202907637451167952258505591137459028985635671875, 25506202907637451167952258505591137459028985635671875, 2648130504479643092610307334825988664408766375652361078125, 908308763036517580765335415845314111892206866848759849796875, 908308763036517580765335415845314111892206866848759849796875, 135226283714587627871600840704802829036176081715842820158209359375, 135226283714587627871600840704802829036176081715842820158209359375, 135226283714587627871600840704802829036176081715842820158209359375, 27772638923018292424641509063111700224620807486718082561272880019078125, 6303861355385615032837554368654156828684655504142757097840279579610371890625, 6303861355385615032837554368654156828684655504142757097840279579610371890625, 6303861355385615032837554368654156828684655504142757097840279579610371890625, 1895968252829843734121321364579530170265683043392488053017736007202354280940046875, 1895968252829843734121321364579530170265683043392488053017736007202354280940046875, 678587893338582200723096250918024222769960883743648791543630912073801823045533117078125, 263982226502895231978696734243377029069301873111303021939928666522214163795704156606380953125, 263982226502895231978696734243377029069301873111303021939928666522214163795704156606380953125, 263982226502895231978696734243377029069301873111303021939928666522214163795704156606380953125, 130153532972760962279544659154620367035299526216923730634240489813445949103670181669053458747796875] The limit of the sequence Theta(3,n) is (1-1/2^3)*Zeta(3) approximately 1.051799790. (Maple12, 10 digits). The values of Theta(3,10^k), k=0..4 are 1., 1.051176344, 1.051793540, 1.051799728, 1.051799789]. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s=4: Theta(4,n), n=1..40: [1, 82/81, 51331/50625, 123296356/121550625, 9988505461/9845600625, 146251554055126/144149438750625, 4177234784807204311/4117052120156600625, 4177316109293528392/4117052120156600625, 348897735816424941428857/343860310127599440800625, 45469045689642442391390873722/44812219476138886724578250625, 45469276109166591994111574347/44812219476138886724578250625, 12724212507884764409110901655089452/12540296310422182199892702233150625, 7952652881902074431185833362754480541/7837685194013863874932938895719140625, 644166078021050808054824782971269564446/634852500715122973869568050553250390625, 455607062681307551694798396892753062061321951/449019111558292892081441960363353489531640625, 420763139153615389741623991738752563981423641150096/414678578925426209987945360676724578005750281640625, 420763488821577189411424921096914428662610672790721/414678578925426209987945360676724578005750281640625, 60109109308403231537976272759342454095213484534478/59239796989346601426849337239532082572250040234375, 112654207650343298169046576180259252641621971638662057333/111024917165350815876745380730178687405690707655696484375, 112654255641589742470334517712107581679919211849546041708/111024917165350815876745380730178687405690707655696484375, 318334113100951437623530813850063562294117597685892728059324163/313729880954078886822687903797490457902191979755868553383984375, 1088321297933526829977651592609314954325149439278107500450498891772938/1072580030735685852348292228190764174966471842581343166177719165234375, 1088321559498539659896469418743679396517296969586741569974819252137313/1072580030735685852348292228190764174966471842581343166177719165234375, 5310663108355424241829116138016804412041095160321843272259462160807820871528/5233848396960342275672766968350136320064568287279171202477258833930040234375, 89256330121162842522049580799773703911006523282468832935672951024332129731911721/87965290007712472627232194437060741131325199204301030400035289221862186219140625, 89256343123775774157410101090626028485064430996548459586977236002012648294802346/87965290007712472627232194437060741131325199204301030400035289221862186219140625, 704275567512923401961808039095858392927600807879275890251342128141685984992006529009051/694088449465345118728195712793933473742639989142752398651900848934608364980590937890625, 704275643364403433691347685644122887859200970998187346327008943457672481499989861359676/694088449465345118728195712793933473742639989142752398651900848934608364980590937890625, 704275709117376502313178881775661033216848601255136842897954574468966335546457361750301/694088449465345118728195712793933473742639989142752398651900848934608364980590937890625, 8533963704994691916791242296247718547055019526393535372547200392342697319271921104026926966286/8410520308101843793217408330576410511323589841478811348080995922747114952089578387749281640625, 118159912969647718225720564243729138191685329937930035674975649708510527340880207460479678243089957151/116450726913249140967725114177236418290515124494330826794525124668004836835354920113812901460312890625, 118159920361954911346375212453999840774927825778243770618573034005032601381743816235675662909902847776/116450726913249140967725114177236418290515124494330826794525124668004836835354920113812901460312890625, 118159926885576740455952102211367673959279538422205788437603280538424270749333849377648009189817498401/116450726913249140967725114177236418290515124494330826794525124668004836835354920113812901460312890625, 2381055100473136964962626949590751750678409342085142424037371450651857297211423903559679842406025838508748146/2346612688566839942786685870524306510578783426015324304766517924725050295654494073158777548687841756844140625, 2381055203998092276527776435786456556961770629069202994205250501834313363204555349520288300744318941009138771/2346612688566839942786685870524306510578783426015324304766517924725050295654494073158777548687841756844140625, 60506617634002134106527585212208288670177354999859339428312979044221410764841593943347142485324244179023809377524676/59631373072412883804153512388970979793051169789988522344273532983894990822127689603501517417217403303402868281640625, 1718281569296615471284373842157839512824245833079730277174556546599282264154956733755002102461094266556168586843874014135541/1693426103671291525775348245818483626169197245028013444766564759598099068059570262133430535479837368404230773553286344140625, 1718281622817242945340007249378722343630641919414853082998191344159847505773890242798210387171985264437448378384994758535702/1693426103671291525775348245818483626169197245028013444766564759598099068059570262133430535479837368404230773553286344140625, 1718281670990204814308655673716982177779375343905702822797756473292338763019573996291123336962887720923001602762674446426327/1693426103671291525775348245818483626169197245028013444766564759598099068059570262133430535479837368404230773553286344140625, 66927211959309931395203623267734272718546695943727226337914706019575691097151279800603215689125306203693675594966767018221941323112/65959083905511202303563401977837848536463952398817970942746723478091486146944774535288352164813039366171629372593160920471219140625] Numerator Theta(4,n), n=1..40: A120269 [1, 82, 51331, 123296356, 9988505461, 146251554055126, 4177234784807204311, 4177316109293528392, 348897735816424941428857, 45469045689642442391390873722, 45469276109166591994111574347, 12724212507884764409110901655089452, 7952652881902074431185833362754480541, 644166078021050808054824782971269564446, 455607062681307551694798396892753062061321951, 420763139153615389741623991738752563981423641150096, 420763488821577189411424921096914428662610672790721, 60109109308403231537976272759342454095213484534478, 112654207650343298169046576180259252641621971638662057333, 112654255641589742470334517712107581679919211849546041708, 318334113100951437623530813850063562294117597685892728059324163, 1088321297933526829977651592609314954325149439278107500450498891772938, 1088321559498539659896469418743679396517296969586741569974819252137313, 5310663108355424241829116138016804412041095160321843272259462160807820871528, 89256330121162842522049580799773703911006523282468832935672951024332129731911721, 89256343123775774157410101090626028485064430996548459586977236002012648294802346, 704275567512923401961808039095858392927600807879275890251342128141685984992006529009051, 704275643364403433691347685644122887859200970998187346327008943457672481499989861359676, 704275709117376502313178881775661033216848601255136842897954574468966335546457361750301, 8533963704994691916791242296247718547055019526393535372547200392342697319271921104026926966286, 118159912969647718225720564243729138191685329937930035674975649708510527340880207460479678243089957151, 118159920361954911346375212453999840774927825778243770618573034005032601381743816235675662909902847776, 118159926885576740455952102211367673959279538422205788437603280538424270749333849377648009189817498401, 2381055100473136964962626949590751750678409342085142424037371450651857297211423903559679842406025838508748146, 2381055203998092276527776435786456556961770629069202994205250501834313363204555349520288300744318941009138771, 60506617634002134106527585212208288670177354999859339428312979044221410764841593943347142485324244179023809377524676, 1718281569296615471284373842157839512824245833079730277174556546599282264154956733755002102461094266556168586843874014135541, 1718281622817242945340007249378722343630641919414853082998191344159847505773890242798210387171985264437448378384994758535702, 1718281670990204814308655673716982177779375343905702822797756473292338763019573996291123336962887720923001602762674446426327, 66927211959309931395203623267734272718546695943727226337914706019575691097151279800603215689125306203693675594966767018221941323112]. Denominators Theta(4,n), n=1..40: A128493 [1, 81, 50625, 121550625, 9845600625, 144149438750625, 4117052120156600625, 4117052120156600625, 343860310127599440800625, 44812219476138886724578250625, 44812219476138886724578250625, 12540296310422182199892702233150625, 7837685194013863874932938895719140625, 634852500715122973869568050553250390625, 449019111558292892081441960363353489531640625, 414678578925426209987945360676724578005750281640625, 414678578925426209987945360676724578005750281640625, 59239796989346601426849337239532082572250040234375, 111024917165350815876745380730178687405690707655696484375, 111024917165350815876745380730178687405690707655696484375, 313729880954078886822687903797490457902191979755868553383984375, 1072580030735685852348292228190764174966471842581343166177719165234375, 1072580030735685852348292228190764174966471842581343166177719165234375, 5233848396960342275672766968350136320064568287279171202477258833930040234375, 87965290007712472627232194437060741131325199204301030400035289221862186219140625, 87965290007712472627232194437060741131325199204301030400035289221862186219140625, 694088449465345118728195712793933473742639989142752398651900848934608364980590937890625, 694088449465345118728195712793933473742639989142752398651900848934608364980590937890625, 694088449465345118728195712793933473742639989142752398651900848934608364980590937890625, 8410520308101843793217408330576410511323589841478811348080995922747114952089578387749281640625, 116450726913249140967725114177236418290515124494330826794525124668004836835354920113812901460312890625, 116450726913249140967725114177236418290515124494330826794525124668004836835354920113812901460312890625, 116450726913249140967725114177236418290515124494330826794525124668004836835354920113812901460312890625, 2346612688566839942786685870524306510578783426015324304766517924725050295654494073158777548687841756844140625, 2346612688566839942786685870524306510578783426015324304766517924725050295654494073158777548687841756844140625, 59631373072412883804153512388970979793051169789988522344273532983894990822127689603501517417217403303402868281640625, 1693426103671291525775348245818483626169197245028013444766564759598099068059570262133430535479837368404230773553286344140625, 1693426103671291525775348245818483626169197245028013444766564759598099068059570262133430535479837368404230773553286344140625, 1693426103671291525775348245818483626169197245028013444766564759598099068059570262133430535479837368404230773553286344140625, 65959083905511202303563401977837848536463952398817970942746723478091486146944774535288352164813039366171629372593160920471219140625] The limit of the sequence Theta(4,n) is (1-1/2^4)*Zeta(4) = (Pi^4)/96 approximately 1.014678032 (Maple12 10 digits). The values of Theta(4,10^k), k=0..4 are [1., 1.014657302, 1.014678011, 1.014678032, 1.014678032]. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k=5: Theta(5,n), n=1..40: [1, 244/243, 762743/759375, 12820180976/12762815625, 3115356499043/3101364196875, 501734380891571068/499477805270915625, 186290962962179367466549/185452612752454075153125, 186291207179611798681792/185452612752454075153125, 264507060005034822095008296869/263316190384861185784690603125, 654945930087597102815813733559637156/651996955695764397260286617707209375, 654946089730308117005814730177159031/651996955695764397260286617707209375, 4215458332009996232497953858159263996273008/4196476041813743307955464949873473110315625, 13173308630403571606953964352496483947864545301/13113987630667947837360827968354603469736328125, 3201114219274599944520158260700693226803242711268/3186698994252311324478681196310168643145927734375, 65658533904258985629756024067493959488420748079909661307/65362857889259301170769577341016119254693952502998046875, 1879748146946507931960416476957496098797999603006646262915007232/1871283128303145811372439015902128971576640624995159036689453125, 1879748194762129160119433849606315951000045244317585059291960357/1871283128303145811372439015902128971576640624995159036689453125, 1879748230390772050403640408729329284056585430223755107393994732/1871283128303145811372439015902128971576640624995159036689453125, 130349182330326918563137684518828050414589617216903199540818589443487649/129762176783878826108540522103839547613464789704117433448354859853515625, 130349183768546134938545196959481827832325934426710622759033399511847024/129762176783878826108540522103839547613464789704117433448354859853515625, 15101761364636795271092041095062837605028895369918595383323181547912710547311449/15033752835670598837275118545507377500054646982382959897196723762515736806640625, 2220086439806563368983602708352321288984524378396288726432288982361701239985670760204532/2210088596818769595945425540015464211296266067794767364218330425028540230942030341796875, 2220086451783537593475232891802272563433686051156287061617060215572521677404303995338907/2210088596818769595945425540015464211296266067794767364218330425028540230942030341796875, 509165745034989188679110037842447983692070882734507082035341470678231918140482072583159935084224/506872784708020890713490792092825419647991620392253395710024540856483534465181565333616689453125, 8557548706961496370111291708897227884559474122334224387603373858504450574936861123231138287817974643/8519010892587707110221639742704116828023795163932602821698382458174918763756306568562095699638671875, 8557548731652474401129975353653535134384840753322618810748370923366762516780557553891242549468365268/8519010892587707110221639742704116828023795163932602821698382458174918763756306568562095699638671875, 3578728319223942129438136910320629219421006558586039125544591498942610928693884802820728814919595600794008999/3562611960098086220698743971468541290025007234311810655793346149399012032644019157298963912223574306630859375, 3578728326302667671231653191645166607390862149721780691755167092101739203635883613421632443805280582020724624/3562611960098086220698743971468541290025007234311810655793346149399012032644019157298963912223574306630859375, 3578728332223656520346685170035973204773850853568019490137394457608321168368404642270811945769295892635958999/3562611960098086220698743971468541290025007234311810655793346149399012032644019157298963912223574306630859375, 2558519847789048709123719352860407919674270065042675217116624802085460377261654695267825155788903793989916556183676076/2546997858182140262574608823982622882323683989460299980513788284723437948730590512574532508372736392442478190439453125, 2160916402024711426216858579362623122290616504133756884981373173960025205240696013493077843468589389621485947093869680811248001/2151184969675558250033683349257683374688541782191092349892317664774550499500884392466144143375864686305001434918335673837890625, 2160916404192290789198415224877724181258457609404849046075941363285876807598057515346687432355064062063958358455894098848357376/2151184969675558250033683349257683374688541782191092349892317664774550499500884392466144143375864686305001434918335673837890625, 2160916406046296090392293482778569798499985674572609737231308019715008659606730342036883599773157389652350004992727433098848001/2151184969675558250033683349257683374688541782191092349892317664774550499500884392466144143375864686305001434918335673837890625, 2917507496082495925670135160445502555764444973069353682939753973760001840832013880463578460556423439808084429597258094752753172764751732/2904368837360004837611459485520648136823488565354719253345246825631730124015534986972982855452762650815061496954271487902299180732421875, 2917507497939472027342893151699461420958935481215484560368042179602893053642179953429891482312218729466797434597940786921905779629986107/2904368837360004837611459485520648136823488565354719253345246825631730124015534986972982855452762650815061496954271487902299180732421875, 5263852662449336290535527203164523312106926097747774964858072316190799997526968647617558320105585233088687162288169148878810840367258034704048432/5240147502494666081520583937723712885000401975825866203250299259144346606659698203464058311827664768637098025674893723295769602260689224189453125, 10912343429501284368107862283679696921438854356791655534429709198685160701961975561437285484895747819184105774440482368194544276081589921306660190664845301/10863200930551588887620944806067110264382409129665675940576959662881090436812164963554473482745467743388485242983006831058462299519543411268363723642578125, 10912343434079028299336186160603090946711579011412061686190059883109360987883883630283234835792776974052977778238757631081210529169600371803784656977110508/10863200930551588887620944806067110264382409129665675940576959662881090436812164963554473482745467743388485242983006831058462299519543411268363723642578125, 10912343438092351970696653964471999436878684073414391534194145101558849519776857332512630943328245116201361035304084559172509694793637345772057891283751133/10863200930551588887620944806067110264382409129665675940576959662881090436812164963554473482745467743388485242983006831058462299519543411268363723642578125, 33577896215130932914943988456888229329639061679181548378649197433663122452788546343930226756048056474993364289580183546621871932312761078812829741041476472163728192/33426681936976521186271320101934615662456473795473788883612677482633142104690577362148894415157757608041628459837931017769153361836865679201607302093862608388671875] Numerator Theta(5,n), n=1..40: A164656 [1, 244, 762743, 12820180976, 3115356499043, 501734380891571068, 186290962962179367466549, 186291207179611798681792, 264507060005034822095008296869, 654945930087597102815813733559637156, 654946089730308117005814730177159031, 4215458332009996232497953858159263996273008, 13173308630403571606953964352496483947864545301, 3201114219274599944520158260700693226803242711268, 65658533904258985629756024067493959488420748079909661307, 1879748146946507931960416476957496098797999603006646262915007232, 1879748194762129160119433849606315951000045244317585059291960357, 1879748230390772050403640408729329284056585430223755107393994732, 130349182330326918563137684518828050414589617216903199540818589443487649, 130349183768546134938545196959481827832325934426710622759033399511847024, 15101761364636795271092041095062837605028895369918595383323181547912710547311449, 2220086439806563368983602708352321288984524378396288726432288982361701239985670760204532, 2220086451783537593475232891802272563433686051156287061617060215572521677404303995338907, 509165745034989188679110037842447983692070882734507082035341470678231918140482072583159935084224, 8557548706961496370111291708897227884559474122334224387603373858504450574936861123231138287817974643, 8557548731652474401129975353653535134384840753322618810748370923366762516780557553891242549468365268, 3578728319223942129438136910320629219421006558586039125544591498942610928693884802820728814919595600794008999, 3578728326302667671231653191645166607390862149721780691755167092101739203635883613421632443805280582020724624, 3578728332223656520346685170035973204773850853568019490137394457608321168368404642270811945769295892635958999, 2558519847789048709123719352860407919674270065042675217116624802085460377261654695267825155788903793989916556183676076, 2160916402024711426216858579362623122290616504133756884981373173960025205240696013493077843468589389621485947093869680811248001, 2160916404192290789198415224877724181258457609404849046075941363285876807598057515346687432355064062063958358455894098848357376, 2160916406046296090392293482778569798499985674572609737231308019715008659606730342036883599773157389652350004992727433098848001, 2917507496082495925670135160445502555764444973069353682939753973760001840832013880463578460556423439808084429597258094752753172764751732, 2917507497939472027342893151699461420958935481215484560368042179602893053642179953429891482312218729466797434597940786921905779629986107, 5263852662449336290535527203164523312106926097747774964858072316190799997526968647617558320105585233088687162288169148878810840367258034704048432, 10912343429501284368107862283679696921438854356791655534429709198685160701961975561437285484895747819184105774440482368194544276081589921306660190664845301, 10912343434079028299336186160603090946711579011412061686190059883109360987883883630283234835792776974052977778238757631081210529169600371803784656977110508, 10912343438092351970696653964471999436878684073414391534194145101558849519776857332512630943328245116201361035304084559172509694793637345772057891283751133, 33577896215130932914943988456888229329639061679181548378649197433663122452788546343930226756048056474993364289580183546621871932312761078812829741041476472163728192] Denominator Theta(5,n), n=1..40: A164657 [1, 243, 759375, 12762815625, 3101364196875, 499477805270915625, 185452612752454075153125, 185452612752454075153125, 263316190384861185784690603125, 651996955695764397260286617707209375, 651996955695764397260286617707209375, 4196476041813743307955464949873473110315625, 13113987630667947837360827968354603469736328125, 3186698994252311324478681196310168643145927734375, 65362857889259301170769577341016119254693952502998046875, 1871283128303145811372439015902128971576640624995159036689453125, 1871283128303145811372439015902128971576640624995159036689453125, 1871283128303145811372439015902128971576640624995159036689453125, 129762176783878826108540522103839547613464789704117433448354859853515625, 129762176783878826108540522103839547613464789704117433448354859853515625, 15033752835670598837275118545507377500054646982382959897196723762515736806640625, 2210088596818769595945425540015464211296266067794767364218330425028540230942030341796875, 2210088596818769595945425540015464211296266067794767364218330425028540230942030341796875, 506872784708020890713490792092825419647991620392253395710024540856483534465181565333616689453125, 8519010892587707110221639742704116828023795163932602821698382458174918763756306568562095699638671875, 8519010892587707110221639742704116828023795163932602821698382458174918763756306568562095699638671875, 3562611960098086220698743971468541290025007234311810655793346149399012032644019157298963912223574306630859375, 3562611960098086220698743971468541290025007234311810655793346149399012032644019157298963912223574306630859375, 3562611960098086220698743971468541290025007234311810655793346149399012032644019157298963912223574306630859375, 2546997858182140262574608823982622882323683989460299980513788284723437948730590512574532508372736392442478190439453125, 2151184969675558250033683349257683374688541782191092349892317664774550499500884392466144143375864686305001434918335673837890625, 2151184969675558250033683349257683374688541782191092349892317664774550499500884392466144143375864686305001434918335673837890625, 2151184969675558250033683349257683374688541782191092349892317664774550499500884392466144143375864686305001434918335673837890625, 2904368837360004837611459485520648136823488565354719253345246825631730124015534986972982855452762650815061496954271487902299180732421875, 2904368837360004837611459485520648136823488565354719253345246825631730124015534986972982855452762650815061496954271487902299180732421875, 5240147502494666081520583937723712885000401975825866203250299259144346606659698203464058311827664768637098025674893723295769602260689224189453125, 10863200930551588887620944806067110264382409129665675940576959662881090436812164963554473482745467743388485242983006831058462299519543411268363723642578125, 10863200930551588887620944806067110264382409129665675940576959662881090436812164963554473482745467743388485242983006831058462299519543411268363723642578125, 10863200930551588887620944806067110264382409129665675940576959662881090436812164963554473482745467743388485242983006831058462299519543411268363723642578125, 33426681936976521186271320101934615662456473795473788883612677482633142104690577362148894415157757608041628459837931017769153361836865679201607302093862608388671875]. The limit of the sequence Theta(5,n) is (1-1/2^5)*Zeta(5) approximately 1.004523763 (Maple12 10 digits). The values of Theta(5,10^k), k=0..4 are [1., 1.004522988, 1.004523763, 1.004523763, 1.004523763]. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e.o.f. ##########################################################