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%I
%S 1,120,83160,81681600,93699005400,117386113965120,155667030019300800,
%T 214804163196079142400,305240072216678400087000,
%U 443655767845074392936328000,656486312795713480715743268160
%N (6*n)!/((n!)^3*(3*n)!).
%C Self-convolution of A092870, where A092870(n) = (12^n/n!^2) * Product_{k=0..n-1} (12k+1)*(12k+5). - Paul D. Hanna, Jan 25 2011
%D M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998. (See Eq. 31.)
%H Vincenzo Librandi, <a href="/A001421/b001421.txt">Table of n, a(n) for n = 0..75</a>
%H R. S. Maier, <a href="http://arxiv.org/abs/0807.1081">Nonlinear differential equations satisfied by certain classical modular forms</a>, p. 34 equation (7.29b)
%F o.g.f.: Hypergeometric2F1(5/12, 1/12; 1; 1728x)^2 [From Jacob Lewis (jacobml(AT)uw.edu), Jul 28 2009]
%F a(n) = C(2n,n) * (12^n/n!^2) * Product_{k=0..n-1} (6k+1)*(6k+5).- Paul D. Hanna, Jan 25 2011
%F G.f.: A(x) = 1 + 120*x + 83160*x^2 + 81681600*x^3 +... - Paul D. Hanna, Jan 25 2011
%F A(x)^(1/2) = 1 + 60*x + 39780*x^2 + 38454000*x^3 +...+ A092870(n)*x^n +... - Paul D. Hanna, Jan 25 2011
%F G.f.: F(1/6, 1/2, 5/6; 1, 1; 1728*x) hypergeometric series. - Michael Somos Feb 28 2011
%p f := n->(6*n)!/( (n!)^3*(3*n)!);
%t Factorial[6 n]/(Factorial[3n] Factorial[n]^3) [From Jacob Lewis (jacobml(AT)uw.edu), Jul 28 2009]
%t a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/6, 1/2, 5/6}, {1, 1}, 1728 x], {x, 0, n}] (* Michael Somos Jul 11 2011 *)
%o (PARI) {a(n)=(2*n)!/n!^2*(12^n/n!^2)*prod(k=0, n-1, (6*k+1)*(6*k+5))} \\ Paul D. Hanna, Jan 25 2011
%o (MAGMA) [Factorial(6*n)/(Factorial(n)^3*Factorial(3*n)): n in [0..15]]; // Vincenzo Librandi, Oct 26 2011
%Y Cf. A092870; variants: A184423, A008977, A184892, A184896, A184898 - Paul D. Hanna, Jan 25 2011
%K nonn
%O 0,2
%A _N. J. A. Sloane_, KUPK78A(AT)prodigy.com (Glenn K Painter)
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