A130416 W. Lang, June 1, 2007.

 Rationals r(n)= A130416(n)/A130417(n), n>=1: 

 r(n):=2*sum(1/((k^4)*binomial(2*k,k)),k=1..n) 

 r(n) tends, in the limit n->infinity, to 

 (18/17)*Zeta(4) = (17/1680)*Pi^4, approximately 1.022194166.

 See the exercise given in A. Comtet's book on p.89.

 r(n), n=1..20:

[1, 49/48, 6623/6480, 741857/725760, 13247611/12960000, 3060203141/2993760000, 13645449045719/13349175840000, 218327192834879/213586813440000, 100212182125865461/98036347368960000, 1904031462407822767/1862690600010240000, 2534265876944902342877/2479241188613629440000, 58288115171766608401171/57022547338113477120000, 128058989033214718801833487/125278536501835309232640000, 2613448755783210204740863/2556704826568067535360000, 19944740504666310380612533/19511694729072094348800000, 187959234515984644745985836617/183878211126775417143091200000, 29788507070227155797522177711591/29141730685995084658838937600000, 2770331157531131178814375090889963/2710180953797542873272021196800000, 703062952151723362591125475535442936029/687797852997601823007593355387494400000, 703062952151723424961017819336159234653/687797852997601823007593355387494400000]


Numerators A130416: 
 
 [1, 49, 6623, 741857, 13247611, 3060203141, 13645449045719, 218327192834879, 100212182125865461, 1904031462407822767, 2534265876944902342877, 58288115171766608401171, 128058989033214718801833487, 2613448755783210204740863, 19944740504666310380612533, 187959234515984644745985836617, 29788507070227155797522177711591, 2770331157531131178814375090889963, 703062952151723362591125475535442936029, 703062952151723424961017819336159234653]


Denominators  A130417:


[1, 48, 6480, 725760, 12960000, 2993760000, 13349175840000, 213586813440000, 98036347368960000, 1862690600010240000, 2479241188613629440000, 57022547338113477120000, 125278536501835309232640000, 2556704826568067535360000, 19511694729072094348800000, 183878211126775417143091200000, 29141730685995084658838937600000, 2710180953797542873272021196800000, 687797852997601823007593355387494400000, 687797852997601823007593355387494400000]



The values for r(10^k), k=0,..,4 are (maple10, 10 digits)

[1., 1.022194165, 1.022194165, 1.022194165]

This should be compared with the value for (17/1680)*Pi^4 (maple10, 10digits)

1.022194166.


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