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A092870 Expansion of Hypergeometric function F(1/12, 5/12; 1; 1728*x) in powers of x.

%I A092870 #32 Aug 20 2018 01:46:29
%S A092870 1,60,39780,38454000,43751038500,54538294552560,72081445966966800,
%T A092870 99225259048241726400,140744828381240373790500,
%U A092870 204278086466816584003782000,301931182921413583820949947280
%N A092870 Expansion of Hypergeometric function F(1/12, 5/12; 1; 1728*x) in powers of x.
%C A092870 Self-convolution yields Sum_{k=0..n} a(n-k)*a(k) = A001421(n). - _Paul D. Hanna_, Jan 25 2011
%H A092870 Seiichi Manyama, <a href="/A092870/b092870.txt">Table of n, a(n) for n = 0..310</a> (terms 0..100 from Vincenzo Librandi)
%H A092870 R. S. Maier, <a href="http://arxiv.org/abs/0807.1081">Nonlinear differential equations satisfied by certain classical modular forms</a>, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.29a).
%F A092870 G.f.: F(1/12, 5/12; 1; 1728*x). a(n) * n^2 = a(n-1) * 12 * (12*n - 7) * (12*n - 11).
%F A092870 G.f. A(x) = y satisfies 0 = (1728*x^2 - x) * y" + (2592*x - 1) * y' + 60 * y.
%F A092870 a(n) = (12^n/n!^2) * Product_{k=0..n-1} (12k+1)*(12k+5). - _Paul D. Hanna_, Jan 25 2011
%F A092870 G.f.: A(x) = 1 + 60*x + 39780*x^2 + 38454000*x^3 +... with  A(x)^2 = 1 + 120*x + 83160*x^2 + 81681600*x^3 +...+ A184894(n)*x^n +... - _Paul D. Hanna_, Jan 25 2011
%F A092870 a(n) ~ 1728^n * GAMMA(11/12) * GAMMA(7/12) / (4*Pi^2*n^(3/2)). - _Vaclav Kotesovec_, Apr 20 2014
%t A092870 CoefficientList[ Series[ Hypergeometric2F1[ 1/12, 5/12, 1, 1728 x], {x, 0, 10}], x]
%o A092870 (PARI) {a(n) = local(an); if( n<1, n==0, an = vector(n+1); an[1] = 1; for(k=1, n, an[k+1] = an[k] * 12 * (12*k - 7) * (12*k - 11) / k^2); an[n+1])}
%o A092870 (PARI) {a(n)=(12^n/n!^2)*prod(k=0, n-1, (12*k+1)*(12*k+5))} \\ _Paul D. Hanna_, Jan 25 2011
%Y A092870 Cf. A001421; variants: A184424, A178529, A184891, A184895, A184897. - _Paul D. Hanna_, Jan 25 2011
%Y A092870 Cf. A289307.
%K A092870 nonn
%O A092870 0,2
%A A092870 _Michael Somos_, Mar 08 2004