A061163 a(n) = (10n)!*n!/((5n)!*(4n)!*(2n)!).

1, 630, 1385670, 3528923580, 9540949030470, 26651569523959380, 75998432812419471900, 219813190240007470094520, 642409325786050322446410310, 1892390644737640220059489996260

Offset

0,2

Comments

According to page 781 of the cited reference the generating function F(x) for a(n) is algebraic but not obviously so and the minimal polynomial satisfied by F(x) is quite large.

This sequence is the particular case a = 5, b = 1 of the following result (see Bober, Theorem 1.2): let a, b be nonnegative integers with a > b and GCD(a,b) = 1. Then (2*a*n)!*(b*n)!/((a*n)!*(2*b*n)!*((a-b)*n)!) is an integer for all integer n >= 0. Other cases include A061162 (a = 3, b = 1), A211419 (a = 3, b = 2) and A211420(a = 4, b = 1) and A211421 (a = 4, b = 3). The o.g.f. Sum_{n >= 1} a(n)*z^n is algebraic over the field of rational functions Q(z) (see Rodriguez-Villegas). - Peter Bala, Apr 10 2012

Continuing the comment above: This is case n = 4 of the array of sequences

A(n, k) = 4^(n*k)*(Gamma((n + 1)*k + 1/2)/Gamma(k + 1/2)) / Gamma(n * k + 1). See the cross-references for other cases. - Peter Luschny, Feb 21 2024

References

M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited - 2001 and Beyond, Springer, Berlin, 2001, pp. 771-808.

Links

Table of n, a(n) for n=0..9.

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, 2007, arXiv:0709.1977 [math.NT], 2007; Jour. of the London Math. Soc., Vol. 79, Issue 2, 422-444.

F. Rodriguez-Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math/0701362 [math.NT], 2007.

Formula

n*(4*n-3)*(2*n-1)*(4*n-1)*a(n) -10*(10*n-9)*(10*n-7)*(10*n-3)*(10*n-1)*a(n-1)=0. - R. J. Mathar, Oct 26 2014

O.g.f. is a generalized hypergeometric function 4F3([1/10, 3/10, 7/10, 9/10], [1/4, 1/2, 3/4], 5^5*z). - Karol A. Penson, Apr 13 2022

From Karol A. Penson, Feb 21 2024: (Start)

(O.g.f.(z))^2 satisfies the algebraic equation of order 15, in which the powers of (O.g.f.(z))^2 are multiplied by polynomials p(n, z) with integer coefficients, in the form: Sum_{n = 0..15} p(n, z)*(O.g.f.(z))^(2*n) = 0.

Here is the list of orders, in the variable z, of all polynomials p(n, z) for n=0..15: 9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12. For example p(15, z) = 2^50*(5^5*z-1)^12. (End)

Maple

A061163 := n->(10*n)!*n!/((5*n)!*(4*n)!*(2*n)!);
# Alternative:
A := (n, k) -> 4^(n*k)*(GAMMA((n + 1)*k + 1/2)/GAMMA(k + 1/2))/GAMMA(n*k + 1):
seq(A(4, k), k = 0..9);  # Peter Luschny, Feb 21 2024

Mathematica

Table[(10n)! n!/((5n)!(4n)!(2n)!), {n, 0, 10}] (* Harvey P. Dale, Oct 24 2022 *)

Crossrefs

Cf. A061164, A211419, A211421.

Cf. A000012 (n=0), A001448 (n=1), A061162 (n=2), A211420 (n=3), this sequence (n=4).

Sequence in context: A058832 A366489 A225390 * A045168 A270802 A119504

Adjacent sequences: A061160 A061161 A061162 * A061164 A061165 A061166

Keyword

easy,nonn

Author

Richard Stanley, Apr 17 2001

Status

approved