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A061163 a(n) = (10n)!*n!/((5n)!*(4n)!*(2n)!). 7

%I

%S 1,630,1385670,3528923580,9540949030470,26651569523959380,

%T 75998432812419471900,219813190240007470094520,

%U 642409325786050322446410310,1892390644737640220059489996260

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

%C 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 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

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

%H J. W. Bober, <a href="http://arxiv.org/abs/0709.1977">Factorial ratios, hypergeometric series, and a family of step functions</a>, 2007, arXiv:0709.1977v1 [math.NT], Jour. of the London Math. Soc., Vol. 79, Issue 2, 422-444.

%H F. Rodriguez-Villegas, <a href="http://arxiv.org/abs/math/0701362"> Integral ratios of factorials and algebraic hypergeometric functions</a>. arXiv:math.NT/0701362

%F 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

%p A061123 := n->(10*n)!*n!/((5*n)!*(4*n)!*(2*n)!);

%Y Cf. A061163, A061164. A211419, A211420, A211421.

%K easy,nonn

%O 0,2

%A _Richard Stanley_, Apr 17 2001

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Last modified January 22 22:16 EST 2020. Contains 331166 sequences. (Running on oeis4.)