A276018 - OEIS

%I #30 Oct 06 2016 02:29:02

%S 1,3,36,588,11025,223587,4769856,105423552,2391796836,55365667500,

%T 1302200499600,31026810250800,747229013540100,18158991471060300,

%U 444709995209640000,10963583748568324800,271862615765280179025,6775869970094509098675,169647707399403264840900,4264689597367270438867500

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

%H Gheorghe Coserea, <a href="/A276018/b276018.txt">Table of n, a(n) for n = 0..200</a>

%H Robert S. Maier, <a href="http://arxiv.org/abs/math/0611041">On Rationally Parametrized Modular Equations</a>, arXiv:math/0611041 [math.NT], 2006.

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

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

%F From _Vaclav Kotesovec_, Aug 25 2016: (Start)

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

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

%F G.f.: 2F1(1/3,1/3;1;27*x). - _Benedict W. J. Irwin_, Oct 05 2016

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

%t 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 *)

%o (PARI)

%o seq(N) = {

%o   a = vector(N); a[1] = 3;

%o   for (n = 2, N, a[n] = 3*(3*n-2)^2/n^2 * a[n-1]);

%o   concat(1, a);

%o };

%o seq(20)

%Y Cf. A091401.

%K nonn

%O 0,2

%A _Gheorghe Coserea_, Aug 22 2016