%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)!*((ab)*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 RodriguezVillegas).  _Peter Bala_, Apr 10 2012
%D M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited  2001 and Beyond, Springer, Berlin, 2001, pp. 771808.
%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, 422444.
%H F. RodriguezVillegas, <a href="http://arxiv.org/abs/math/0701362"> Integral ratios of factorials and algebraic hypergeometric functions</a>. arXiv:math.NT/0701362
%F n*(4*n3)*(2*n1)*(4*n1)*a(n) 10*(10*n9)*(10*n7)*(10*n3)*(10*n1)*a(n1)=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
