W. Lang, Oct 17 2008

Rationals r(n):= A145559(n)/(2*A145560(n)) = sum(((-1)^(k+1))/(binomial(2*k,k)*k^2),k=1..n), n>=1,

(in lowest terms).

Numerators  A145559(n), n=1..25:

[1, 11, 167, 4667, 7781, 770269, 70095379, 280380781, 14299427671, 271689093997, 229890777659, 
68737342138891, 7770308251333, 893585448657907, 43189963354470841, 5355555455879234209, 
10116049194470941417, 819399984751544533657, 576038189280433285982311, 19863385837255542358043, 
23617565760497054697989927, 92323211609215563526200151, 2123433867011959013790559793, 
2320962598827024698759483527, 499006958747810323870968849569]


Denominators are 2* A145560(n), n=1..25: 

[2, 24, 360, 10080, 16800, 1663200, 151351200, 605404800, 30875644800, 586637251200, 496385366400, 
148419224553600, 16777825384320, 1929449919196800, 93256746094512000, 11563836515719488000, 
21842802307470144000, 1769266986905081664000, 1243794691794272409792000, 42889472130836979648000, 
50995582363565168801472000, 199346367421209296223936000, 4584966450687813813150528000, 
5011474957728540679490112000, 1077467115911636246090374080000]


A145560(n), n=1..25:

[1, 12, 180, 5040, 8400, 831600, 75675600, 302702400, 15437822400, 293318625600, 248192683200, 
74209612276800, 8388912692160, 964724959598400, 46628373047256000, 5781918257859744000, 
10921401153735072000, 884633493452540832000, 621897345897136204896000, 21444736065418489824000, 
25497791181782584400736000, 99673183710604648111968000, 2292483225343906906575264000, 
2505737478864270339745056000, 538733557955818123045187040000]


Rationals r(n), n=1..25:

[1/2, 11/24, 167/360, 4667/10080, 7781/16800, 770269/1663200, 70095379/151351200, 280380781/605404800, 
14299427671/30875644800, 271689093997/586637251200, 229890777659/496385366400, 
68737342138891/148419224553600, 7770308251333/16777825384320, 893585448657907/1929449919196800, 
43189963354470841/93256746094512000, 5355555455879234209/11563836515719488000, 
10116049194470941417/21842802307470144000, 819399984751544533657/1769266986905081664000, 
576038189280433285982311/1243794691794272409792000, 19863385837255542358043/42889472130836979648000, 
23617565760497054697989927/50995582363565168801472000, 
92323211609215563526200151/199346367421209296223936000, 
2123433867011959013790559793/4584966450687813813150528000, 
2320962598827024698759483527/5011474957728540679490112000, 
499006958747810323870968849569/1077467115911636246090374080000]


Some values are r(10^k),k=0..3: (Maple11, 10 digits)  .5000000000, .4631296315, .4631296412, .4631296412 .

This should be compared with the limit 2*(ln(phi))^2, with the golden section phi:=(1 + sqrt(5))/2. 
This is approximately  0.4631296414  (Maple11, 10 digits).

The values for r(10^k)/2, k=0..3 are: [.2500000000, .2315648158, .2315648206, .2315648206 . 
The limit is (ln(phi))^2; approximately 0.2315648207 .


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