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%I #5 Mar 30 2012 18:37:27
%S 1,1,3,12,54,265,1373,7388,40888,231250,1330618,7764670,45841323,
%T 273316120,1643345418,9953021248,60665811025,371850104167,
%U 2290623433302,14173331572490,88049709138896,548978010516319,3434070688405887,21545961024510032
%N G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n^2) * A(x)^(2*n+1).
%F The g.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} (-x)^n*A(x)^(2*n+1) * Product_{k=1..n} (1 + x^(4*k-3)*A(x)^2)/(1 + x^(4*k-1)*A(x)^2);
%F (2) 1 = A(x)/(1+ x*A(x)^2/(1- x*(1-x^2)*A(x)^2/(1+ x^5*A(x)^2/(1- x^3*(1-x^4)*A(x)^2/(1+ x^9*A(x)^2/(1- x^5*(1-x^6)*A(x)^2/(1+ x^13*A(x)^2/(1- x^7*(1-x^8)*A(x)^2/(1- ...))))))))) (continued fraction);
%F due to identities of a partial elliptic theta function.
%e G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 54*x^4 + 265*x^5 + 1373*x^6 +...
%e which satisfies:
%e 1 = A(x) - x*A(x)^3 + x^4*A(x)^5 - x^9*A(x)^7 + x^16*A(x)^9 -+...
%e Related expansions.
%e A(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 270*x^4 + 1398*x^5 + 7518*x^6 +...
%e A(x)^5 = 1 + 5*x + 25*x^2 + 130*x^3 + 695*x^4 + 3816*x^5 +...
%o (PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(#A)+1, (-x)^(m^2)*Ser(A)^(2*m+1) ), #A-1)); if(n<0, 0, A[n+1])}
%Y Cf. A193111, A193112, A193113, A193114, A193116.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 16 2011
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