A000894 a(n) = (2*n)!*(2*n+1)! /((n+1)! *n!^3).

1, 6, 60, 700, 8820, 116424, 1585584, 22084920, 312869700, 4491418360, 65166397296, 953799087696, 14062422446800, 208618354980000, 3111393751416000, 46619049708716400, 701342468412012900

Offset

0,2

Comments

a(n) = (n+1) * A000891(n) = A248045(n+1) / A000142(n). - Reinhard Zumkeller, Sep 30 2014

This sequence is one half of the odd part of the bisection of A241530. The even part is given in A002894. - Wolfdieter Lang, Sep 06 2016

References

E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 96.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..180

Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023). See p. 36.

Pedro J. Miana and Natalia Romero, Moments of combinatorial and Catalan numbers, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1876-1887. See Omega1 Remark 3 p. 1882.

Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013 (see Omega_1).

Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.

Formula

a(n) = C(2*n+1,n)*C(2*n,n) = A001700(n)*A000984(n) = A000984(n)*A000984(n+1)/2, n>=0. - Zerinvary Lajos, Jan 23 2007

G.f.: (EllipticK(4*x^(1/2)) - EllipticE(4*x^(1/2)))/(4*x*Pi). - Mark van Hoeij, Oct 24 2011

n*(n+1)*a(n) -4*(2*n-1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Sep 08 2013

a(n) = A103371(2*n,n) = A132813(2*n,n). - Reinhard Zumkeller, Apr 04 2014

0 = a(n)*(+65536*a(n+2) - 23040*a(n+3) + 1400*a(n+4)) + a(n+1)*(-1536*a(n+2) + 1184*a(n+3) - 90*a(n+4)) + a(n+2)*(-24*a(n+2) - 6*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, May 28 2014

0 = a(n+1)^3 * (+256*a(n) - 6*a(n+1) + a(n+2)) + a(n) * a(n+1) * a(n+

2) * (-768*a(n) - 20*a(n+1) - 3*a(n+2)) + 90*a(n)^2*a(n+2)^2 for all n in Z. - Michael Somos, Sep 17 2014

a(n) = A241530(2n+1)/2, n >= 0. - Wolfdieter Lang, Sep 06 2016

a(n) ~ 2^(4*n+1)/(Pi*n). - Ilya Gutkovskiy, Sep 06 2016

Example

G.f. = 1 + 6*x + 60*x^2 + 700*x^3 + 8820*x^4 + 116424*x^5 + ...

Maple

seq(binomial(2*n+1, n)*binomial(2*n, n), n=0..16); # Zerinvary Lajos, Jan 23 2007

Mathematica

a[ n_] := Binomial[2 n + 1, n] Binomial[2 n, n]; (* Michael Somos, May 28 2014 *)
a[ n_] := SeriesCoefficient[ (EllipticK[ 16 x] - EllipticE[ 16 x]) / (4 x Pi), {x, 0, n}]; (* Michael Somos, May 28 2014 *)
Table[(2 n)!*(2 n + 1)!/((n + 1)!*n!^3), {n, 0, 16}] (* Michael De Vlieger, Sep 06 2016 *)

Prog

(Magma) [Factorial(2*n)*Factorial(2*n+1) /(Factorial(n+1)* Factorial(n)^3): n in [0..20]]; // Vincenzo Librandi, Oct 25 2011
(Haskell)
a000894 n = a132813 (2 * n) n  -- Reinhard Zumkeller, Apr 04 2014
(PARI) {a(n) =  binomial( 2*n + 1, n) * binomial( 2*n, n)}; /* Michael Somos, May 28 2014 */

Crossrefs

First differences of A082578. Cf. A002463.

Cf. A001700, A000984.

Cf. A000142, A000891, A248045, A002894, A241530.

Sequence in context: A232969 A232246 A086984 * A112117 A065944 A357771

Adjacent sequences: A000891 A000892 A000893 * A000895 A000896 A000897

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved