A281821 - OEIS

A281821

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

%I #18 Feb 16 2025 08:33:40

%S 16,3456,432000,1185408000,32006016000,42600007296000,

%T 93592216029312000,5989901825875968,735709691763215769600,

%U 25231163879019484818432000,25231163879019484818432000,306987570916030071785862144000

%N Denominator 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).

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

%H Seiichi Manyama, <a href="/A281821/b281821.txt">Table of n, a(n) for n = 1..384</a>

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

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

%t Table[Denominator@ Sum[(30 k - 11)/(4 (2 k - 1)*k^3*Binomial[2 k, k]^2), {k, n}], {n, 12}] (* _Michael De Vlieger_, Feb 02 2017 *)

%Y Cf. A002117, A281820.

%K nonn,frac,changed

%O 1,1

%A _Seiichi Manyama_, Jan 31 2017