A281820 Numerator of Sum_{k=1..n} (30k-11)/(4*(2k-1)*k^3*binomial(2k,k)^2).

19, 4153, 519283, 1424927267, 38473051777, 51207632802437, 112503169355608589, 7200202839028523, 884364913705304409923, 30329294715526225502633653, 30329294715526370166581653, 369016528803809437978645999301

Offset

1,1

Comments

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

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

References

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.

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

Links

Seiichi Manyama, Table of n, a(n) for n = 1..384

Eric Weisstein's World of Mathematics, Apery's Constant

Example

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

Mathematica

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

Crossrefs

Cf. A002117, A281821.

Sequence in context: A125197 A375090 A238563 * A213450 A099809 A271588

Adjacent sequences: A281817 A281818 A281819 * A281821 A281822 A281823

Keyword

nonn,frac

Author

Seiichi Manyama, Jan 31 2017

Status

approved