A002736 Apéry numbers: a(n) = n^2*C(2n,n). (Formerly M2136 N0848)

0, 2, 24, 180, 1120, 6300, 33264, 168168, 823680, 3938220, 18475600, 85357272, 389398464, 1757701400, 7862853600, 34901442000, 153876579840, 674412197580, 2940343837200, 12759640231800, 55138611528000, 237371722628040, 1018383898440480

Offset

0,2

Comments

Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row n-1 of B equals -a(n-1). - T. D. Noe, May 01 2011

References

J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933, p. 93.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Paul S. Bruckman, Problem B-871, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 37, No. 1 (1999), p. 85.

J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].

J. Ser, Les Calculs Formels des Séries de FactoriellesGauthier-Villars, Paris, 1933 (Annotated scans of some selected pages).

Indulis Strazdins, Solution to problem B-871, Fibonacci Quartely, Vol. 38, No. 1 (2000), pp. 86-87.

Hans J. H. Tuenter, Walking into an absolute sum, arXiv:math/0606080 [math.NT], 2006. Published version on Walking into an absolute sum, The Fibonacci Quarterly, Vol. 40, No. 2 (May 2002), pp. 175-180.

A. J. van der Poorten, A proof that Euler missed...Apery's proof of the irrationality of zeta(3), Math. Intelligencer, Vol. 1 (1978/1979), pp. 195-203.

Formula

G.f.: x*(4*x+2)/((1-4*x)^(5/2)). - Marco A. Cisneros Guevara, Jul 25 2011

Sum_{n>=1} 1/a(n) = Pi^2/18 (Euler). - Benoit Cloitre, Apr 07 2002

From Ilya Gutkovskiy, Jan 17 2017: (Start)

a(n) ~ 4^n*n^(3/2)/sqrt(Pi).

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(phi)^2 = A086467, where phi is the golden ratio. (End)

D-finite with recurrence: (-n+1)*a(n) +2*(n+4)*a(n-1) +4*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 21 2020

a(n) = (2n)!/(Gamma(n))^2. - Diego Rattaggi, Mar 30 2020

a(n) = Sum_{k=0..2*n} binomial(2*n,k)*abs(n-k)^3 (Bruckman, 1999; Strazdins, 2000). - Amiram Eldar, Jan 12 2022

Maple

seq(n^2*binomial(2*n, n), n=0..50); # Robert Israel, Aug 07 2014

Mathematica

CoefficientList[ Series[x (4 x + 2)/(1 - 4 x)^(5/2), {x, 0, 20}], x] (* Robert G. Wilson v, Aug 08 2011 *)
Table[n^2 Binomial[2n, n], {n, 0, 30}] (* Harvey P. Dale, Jun 21 2017 *)

Prog

(MuPAD) combinat::catalan(n)*(n+1)*n^2 $ n = 0..36 // Zerinvary Lajos, Apr 17 2007
(Magma) [n^2*Binomial(2*n, n): n in [0..30]]; // Vincenzo Librandi, Aug 08 2014
(PARI) x='x+O('x^100); concat(0, Vec(x*(4*x+2)/((1-4*x)^(5/2)))) \\ Altug Alkan, Mar 21 2016
(PARI) a(n) = n^2*binomial(2*n, n); \\ Michel Marcus, Mar 21 2016
(Sage) [n^2*(n+1)*catalan_number(n) for n in (0..30)] # G. C. Greubel, Mar 23 2022

Crossrefs

Cf. A000108, A005258, A005259, A005429, A005430.

A diagonal of A331431.

Sequence in context: A279853 A052411 A073066 * A309318 A131972 A059387

Adjacent sequences: A002733 A002734 A002735 * A002737 A002738 A002739

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved