A191803 - OEIS

%I #4 Mar 30 2012 18:37:26

%S 1,1,6,61,791,11701,188462,3225915,57840755,1076423857,20666351126,

%T 407645638428,8237858879315,170229866493435,3592746391559133,

%U 77393340642273491,1701286171473636404,38169860244429063080

%N  G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(5*n^2).

%F  Let A = g.f. A(x), then A satisfies:

%F (1) A = Sum_{n>=0} x^n*A^(5*n)*Product_{k=1..n} (1-x*A^(20*k-15))/(1-x*A^(20*k-5));

%F (2) A = 1/(1- A^5*x/(1- A^5*(A^10-1)*x/(1- A^25*x/(1- A^15*(A^20-1)*x/(1- A^45*x/(1- A^25*(A^30-1)*x/(1- A^65*x/(1- A^35*(A^40-1)*x/(1- ...))))))))) (continued fraction);

%F due to a q-series identity and an identity of a partial elliptic theta function, respectively.

%e  G.f.: A(x) = 1 + x + 6*x^2 + 61*x^3 + 791*x^4 + 11701*x^5 + 188462*x^6 +...

%e where the g.f. satisfies:

%e A(x) = 1 + x*A(x)^5 + x^2*A(x)^20 + x^3*A(x)^45 + x^4*A(x)^80 +...+ x^n*A(x)^(5*n^2) +...

%o  (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^(5*m^2)));polcoeff(A,n)}

%Y  Cf. A107595, A191800, A191801, A191802, A191804.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 16 2011