A005429 Apéry numbers: n^3*C(2n,n). (Formerly M2169)

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

Offset

0,2

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.

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)

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 *)

Prog

(Magma) [Binomial(2*n, n)*n^3 : n in [0..30]]; // Wesley Ivan Hurt, Oct 21 2014
(SageMath) [n^3*binomial(2*n, n) for n in range(31)] # G. C. Greubel, Nov 19 2022

Crossrefs

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

Sequence in context: A058090 A051252 A231654 * A035606 A157057 A290690

Adjacent sequences: A005426 A005427 A005428 * A005430 A005431 A005432

Keyword

nonn,nice,easy

Author

Simon Plouffe

Extensions

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

Status

approved