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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Examples

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

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.

Crossrefs

First differences of A082578.

Cf. A000108, A000142, A000217, A000891, A000984, A001700, A002463, A002894, A103371, A132813, A241530, A248845.

Programs

Haskell

a000894 n = a132813 (2 * n) n  -- Reinhard Zumkeller, Apr 04 2014

Magma

[Factorial(2*n)*Factorial(2*n+1) /(Factorial(n+1)* Factorial(n)^3): n in [0..20]]; // Vincenzo Librandi, Oct 25 2011

Magma

A000894:= func< n | Binomial(2*n+2,2)*Catalan(n)^2 >;
[A000894(n): n in [0..40]]; // G. C. Greubel, Mar 12 2025

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 *)

PARI

{a(n) =  binomial( 2*n + 1, n) * binomial( 2*n, n)}; /* Michael Somos, May 28 2014 */

SageMath

def A000894(n): return binomial(2*n+2,2)*catalan_number(n)^2
print([A000894(n) for n in range(41)]) # G. C. Greubel, Mar 12 2025

Formula

From Zerinvary Lajos, Jan 23 2007: (Start)

a(n) = C(2*n+1,n)*C(2*n,n) = A001700(n)*A000984(n).

a(n) = A000984(n)*A000984(n+1)/2, n>=0. (End)

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

n*(n+1)*a(n) = 4*(2*n-1)*(2*n+1)*a(n-1). - 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) = (n+1) * A000891(n) = A248045(n+1) / A000142(n). - Reinhard Zumkeller, Sep 30 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

a(n) = A000217(n+1)*A000108(n)*A000108(n+1) = A000217(2*n+1)*A000108(n)^2. - G. C. Greubel, Mar 12 2025