OFFSET
0,2
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.
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
EXAMPLE
A(x) = 1 + 3*x + 36*x^2 + 588*x^3 + ... is the g.f.
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 *)
PROG
(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)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Aug 22 2016
STATUS
approved