A276018 n^2 * a(n) = 3*(3*n-2)^2 * a(n-1), with a(0) = 1.

1, 3, 36, 588, 11025, 223587, 4769856, 105423552, 2391796836, 55365667500, 1302200499600, 31026810250800, 747229013540100, 18158991471060300, 444709995209640000, 10963583748568324800, 271862615765280179025, 6775869970094509098675, 169647707399403264840900, 4264689597367270438867500

Offset: 0

Author

Gheorghe Coserea, Aug 22 2016

Keywords

• nonn

Examples

			A(x) = 1 + 3*x + 36*x^2 + 588*x^3 + ... is the g.f.
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..200
• Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.

Crossrefs

Cf. A091401.

Programs

Mathematica

Table[FullSimplify[3^(3*n) * Gamma[n + 1/3]^2 / (Gamma[1/3]^2 * Gamma[n+1]^2)], {n, 0, 20}] (* Vaclav Kotesovec, Aug 25 2016 *)

PARI

seq(N) = {
  a = vector(N); a[1] = 3;
  for (n = 2, N, a[n] = 3*(3*n-2)^2/n^2 * a[n-1]);
  concat(1, a);
};
seq(20)

Formula

n^2 * a(n) = 3*(3*n-2)^2 * a(n-1), with a(0) = 1.

0 = 9*x*(x+27)*y'' + (15*x+243)*y' + y, where y(x) = A(x/-729).

From Vaclav Kotesovec, Aug 25 2016: (Start)

a(n) = 3^(3*n) * Gamma(n+1/3)^2 / (Gamma(1/3)^2 * Gamma(n+1)^2).

a(n) ~ 3^(3*n) / (Gamma(1/3)^2 * n^(4/3)). (End)

G.f.: 2F1(1/3,1/3;1;27*x). - Benedict W. J. Irwin, Oct 05 2016