A218295 - OEIS

%I #3 Oct 26 2012 18:12:06

%S 1,2,14,158,2274,37410,670670,12786622,255519106,5302716866,

%T 113586849614,2501007496542,56446396937186,1303401799574242,

%U 30756416720161422,741216834445478270,18240706372460480002,458484823574294544770,11776969626284389958030

%N  G.f. satisfies: A(x) = 1 + Sum_{n>=1} 2*x^n * A(x)^(3*n^2).

%C  Given g.f. A(x), then Q = A(-x^2) satisfies:

%C Q = (1-x)*Sum_{n>=0} x^n*Product_{k=1..n} (1 - x*Q^(3*k))/(1 + x*Q^(3*k))

%C due to a q-series expansion for the Jacobi theta_4 function.

%e  G.f.: A(x) = 1 + 2*x + 14*x^2 + 158*x^3 + 2274*x^4 + 37410*x^5 +...

%e where

%e A(x) = 1 + 2*x*A(x)^3 + 2*x^2*A(x)^12 + 2*x^3*A(x)^27 + 2*x^4*A(x)^48 + ...

%o  (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 2*x^m*(A+x*O(x^n))^(3*m^2))); polcoeff(A, n)}

%o for(n=0,30,print1(a(n),", "))

%Y  Cf. A176719, A218294.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 26 2012