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%I #28 Sep 08 2022 08:45:03
%S 1,5542680,190818980609400,7691041400616850556280,
%T 330014847932376708502470210680,
%U 14647137653300940580784413641872332680,663999280578266939183818080578580843597787800,30541460340748361003270983719744457382865889296237000
%N a(n) = (20*n)!n!/((10*n)!(7*n)!(4*n)!).
%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.
%D M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited - 2001 and Beyond, Springer, Berlin, 2001, pp. 771-808.
%H Vincenzo Librandi, <a href="/A061164/b061164.txt">Table of n, a(n) for n = 0..22</a>
%H J. W. Bober, <a href="http://arxiv.org/abs/0709.1977">Factorial ratios, hypergeometric series, and a family of step functions</a>, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., Vol. 79, Issue 2 (2009), 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, 2007.
%F One of the 52 sporadic integral factorial ratio sequences found by V. I. Vasyunin (see Bober, Table 2, Entry 43). The o.g.f. sum {n >= 1} a(n)*z^n is an algebraic function over the field of rational functions Q(z) (see Rodriguez-Villegas). - _Peter Bala_, Apr 10 2012
%F O.g.f. is a generalized hypergeometric function 8F7([1/20, 3/20, 7/20, 9/20, 11/20, 13/20, 17/20, 19/20], [1/7, 2/7, 3/7, 1/2, 4/7, 5/7, 6/7], ((2^22)*(5^10)*x)/7^7). - _Karol A. Penson_, Apr 11 2022
%p A061164 := proc(n)
%p binomial(20*n,10*n)*binomial(10*n,3*n)/binomial(4*n,n) ;
%p end proc:
%p seq(A061164(n),n=0..10) ; # _R. J. Mathar_, oct 26 2011
%t Table[((20n)!n!)/((10n)!(7n)!(4n)!),{n,0,10}] (* _Harvey P. Dale_, Oct 25 2011 *)
%o (Magma) [Factorial(20*n)*Factorial(n)/(Factorial(10*n)*Factorial(7*n)*Factorial(4*n)): n in [0..8]]; // _Vincenzo Librandi_, Oct 26 2011
%o (PARI) a(n)=(20*n)!*n!/(10*n)!/(7*n)!/(4*n)! \\ _Charles R Greathouse IV_, Apr 10 2012
%Y Cf. A061162, A061163.
%K easy,nonn
%O 0,2
%A _Richard Stanley_, Apr 17 2001
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