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

1, 60, 39780, 38454000, 43751038500, 54538294552560, 72081445966966800, 99225259048241726400, 140744828381240373790500, 204278086466816584003782000, 301931182921413583820949947280

Offset

0,2

Comments

Self-convolution yields Sum_{k=0..n} a(n-k)*a(k) = A001421(n). - Paul D. Hanna, Jan 25 2011

Links

Seiichi Manyama, Table of n, a(n) for n = 0..310 (terms 0..100 from Vincenzo Librandi)

R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.29a).

Formula

G.f.: F(1/12, 5/12; 1; 1728*x). a(n) * n^2 = a(n-1) * 12 * (12*n - 7) * (12*n - 11).

G.f. A(x) = y satisfies 0 = (1728*x^2 - x) * y" + (2592*x - 1) * y' + 60 * y.

a(n) = (12^n/n!^2) * Product_{k=0..n-1} (12k+1)*(12k+5). - Paul D. Hanna, Jan 25 2011

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

a(n) ~ 1728^n * GAMMA(11/12) * GAMMA(7/12) / (4*Pi^2*n^(3/2)). - Vaclav Kotesovec, Apr 20 2014

Mathematica

CoefficientList[ Series[ Hypergeometric2F1[ 1/12, 5/12, 1, 1728 x], {x, 0, 10}], x]

Prog

(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])}
(PARI) {a(n)=(12^n/n!^2)*prod(k=0, n-1, (12*k+1)*(12*k+5))} \\ Paul D. Hanna, Jan 25 2011

Crossrefs

Cf. A001421; variants: A184424, A178529, A184891, A184895, A184897. - Paul D. Hanna, Jan 25 2011

Cf. A289307.

Sequence in context: A123482 A263606 A220371 * A221936 A225990 A185560

Adjacent sequences: A092867 A092868 A092869 * A092871 A092872 A092873

Keyword

nonn

Author

Michael Somos, Mar 08 2004

Status

approved