A218294 - OEIS

%I #8 Nov 07 2024 15:21:05

%S 1,2,10,82,866,10482,138698,1957346,29024642,448005922,7153738058,

%T 117681081522,1988787934818,34465473701522,611806834645642,

%U 11118408274591938,206835953956603394,3939803761941599042,76880490874588995978,1538019374456939130386

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

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

%e G.f.: A(x) = 1 + 2*x + 10*x^2 + 82*x^3 + 866*x^4 + 10482*x^5 + 138698*x^6 +...

%e where

%e A(x) = 1 + 2*x*A(x)^2 + 2*x^2*A(x)^8 + 2*x^3*A(x)^18 + 2*x^4*A(x)^32 + ...

%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))^(2*m^2))); polcoeff(A, n)}

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

%Y Cf. A176719, A218295.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 26 2012