A281820 - OEIS

%I #26 Mar 24 2022 10:34:11

%S 19,4153,519283,1424927267,38473051777,51207632802437,

%T 112503169355608589,7200202839028523,884364913705304409923,

%U 30329294715526225502633653,30329294715526370166581653,369016528803809437978645999301

%N Numerator of Sum_{k=1..n} (30k-11)/(4*(2k-1)*k^3*binomial(2k,k)^2).

%C In 1990, Gosper gave the following combinatorial identity: zeta(3) = Sum_{k>=1} (30k-11)/(4*(2k-1)*k^3*binomial(2k,k)^2).

%C Conjecture: Sum_{n >= k+1} 1/(n^3*(n^2 - 1)^2*(n^2 - 4)^2*...*(n^2 - k^2)^2) = Sum_{n >= k+1} 1/(n*binomial(n,k)^2*binomial(n+k,k)^2*(n-k)^2) = zeta(3) - A281820(k)/A281821(k). - _Peter Bala_, Jan 17 2022

%D Ralph William Gosper Jr, A calculus of series rearrangements in Algorithms and Complexity, New directions and Recent Results, ed. J. F. Traub, Academic Press Inc., 1976, p. 122.

%D Lloyd James Peter Kilford, Modular Forms: A Classical and Computational Introduction, World Scientific, 2008 page 188.

%H Seiichi Manyama, <a href="/A281820/b281820.txt">Table of n, a(n) for n = 1..384</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AperysConstant.html">Apery's Constant</a>

%e 19/16, 4153/3456, 519283/432000, 1424927267/1185408000, ...

%t Table[Sum[(30k-11)/(4(2k-1)k^3 Binomial[2k,k]^2),{k,n}],{n,20}]//Numerator (* _Harvey P. Dale_, Dec 31 2021 *)

%Y Cf. A002117, A281821.

%K nonn,frac

%O 1,1

%A _Seiichi Manyama_, Jan 31 2017