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A002736 Apéry numbers: a(n) = n^2*C(2n,n).
(Formerly M2136 N0848)
14
0, 2, 24, 180, 1120, 6300, 33264, 168168, 823680, 3938220, 18475600, 85357272, 389398464, 1757701400, 7862853600, 34901442000, 153876579840, 674412197580, 2940343837200, 12759640231800, 55138611528000, 237371722628040, 1018383898440480 (list; graph; refs; listen; history; text; internal format)
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

J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)

I. Strazdins, Solution to problem B-871, Fibonacci Quartely, 38.1 (2000), 86-87.

H. J. H. Tuenter, Walking into an absolute sum, arXiv:math/0606080 [math.NT], 2006.

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.

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

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

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

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Last modified October 27 09:08 EDT 2020. Contains 338035 sequences. (Running on oeis4.)