This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A061163 #52 Aug 13 2025 13:24:35
%S A061163 1,630,1385670,3528923580,9540949030470,26651569523959380,
%T A061163 75998432812419471900,219813190240007470094520,
%U A061163 642409325786050322446410310,1892390644737640220059489996260
%N A061163 a(n) = (10n)!*n!/((5n)!*(4n)!*(2n)!).
%C A061163 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.
%C A061163 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
%C A061163 Continuing the comment above: This is case n = 4 of the array of sequences
%C A061163 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
%D A061163 M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited - 2001 and Beyond, Springer, Berlin, 2001, pp. 771-808.
%H A061163 J. W. Bober, <a href="http://arxiv.org/abs/0709.1977">Factorial ratios, hypergeometric series, and a family of step functions</a>, 2007, arXiv:0709.1977 [math.NT], 2007; Jour. of the London Math. Soc., Vol. 79, Issue 2, 422-444.
%H A061163 M. Kontsevich and D. Zagier, <a href="http://www.ihes.fr/~maxim/TEXTS/Periods.pdf">Periods</a>, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22, p. 11.
%H A061163 F. Rodriguez-Villegas, <a href="http://arxiv.org/abs/math/0701362">Integral ratios of factorials and algebraic hypergeometric functions</a>, arXiv:math/0701362 [math.NT], 2007.
%F A061163 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
%F A061163 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
%F A061163 From _Karol A. Penson_, Feb 21 2024: (Start)
%F A061163 (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.
%F A061163 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)
%F A061163 a(n) ~ 5^(5*n) / (2^(3/2) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 27 2024
%p A061163 A061163 := n->(10*n)!*n!/((5*n)!*(4*n)!*(2*n)!);
%p A061163 # Alternative:
%p A061163 A := (n, k) -> 4^(n*k)*(GAMMA((n + 1)*k + 1/2)/GAMMA(k + 1/2))/GAMMA(n*k + 1):
%p A061163 seq(A(4, k), k = 0..9); # _Peter Luschny_, Feb 21 2024
%t A061163 Table[(10n)! n!/((5n)!(4n)!(2n)!),{n,0,10}] (* _Harvey P. Dale_, Oct 24 2022 *)
%Y A061163 Cf. A061164, A211419, A211421.
%Y A061163 Cf. A000012 (n=0), A001448 (n=1), A061162 (n=2), A211420 (n=3), this sequence (n=4).
%K A061163 easy,nonn
%O A061163 0,2
%A A061163 _Richard Stanley_, Apr 17 2001