A005429 Apéry numbers: n^3*C(2n,n).

0, 2, 48, 540, 4480, 31500, 199584, 1177176, 6589440, 35443980, 184756000, 938929992, 4672781568, 22850118200, 110079950400, 523521630000, 2462025277440, 11465007358860, 52926189069600, 242433164404200, 1102772230560000, 4984806175188840, 22404445765690560

Offset: 0

Author

Simon Plouffe

Keywords

• nonn
• nice
• easy

References

• S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.3.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=0..200
• M. Kondratiewa and S. Sadov, Markov's transformation of series and the WZ method, arXiv:math/0405592 [math.CA], 2004.
• A. J. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3), Math. Intelligencer 1 (1978/1979), 195-203.
• I. J. Zucker, On the series Sum(k>=1) C(2k,k)^(-1)*k^(-n) and related sums, J. Number Theory 20 (1985), no. 1, 92-102.

Crossrefs

Cf. A002736, A005258, A005259, A005429, A005430, A145438.

Programs

Magma

[Binomial(2*n,n)*n^3 : n in [0..30]]; // Wesley Ivan Hurt, Oct 21 2014

Mathematica

Table[n^3 Binomial[2n,n],{n,0,30}] (* Harvey P. Dale, Apr 08 2012 *)
CoefficientList[Series[(2*x*(2*x*(2*x+5)+1))/(1-4*x)^(7/2), {x,0,30}], x] (* Vincenzo Librandi, Oct 22 2014 *)

SageMath

[n^3*binomial(2*n,n) for n in range(31)] # G. C. Greubel, Nov 19 2022

Formula

Sum_{n>=1} (-1)^(n+1) / a(n) = 2 * zeta(3) / 5.

G.f.: (2*x*(2*x*(2*x + 5) + 1))/(1-4*x)^(7/2). - Harvey P. Dale, Apr 08 2012

From Ilya Gutkovskiy, Jan 17 2017: (Start)

a(n) ~ 4^n*n^(5/2)/sqrt(Pi).

Sum_{n>=1} 1/a(n) = (1/2)*4F3(1,1,1,1; 3/2,2,2; 1/4) = A145438. (End)

Extensions

Entry revised by N. J. A. Sloane, Apr 06 2004