A193116 - OEIS

%I #7 May 14 2012 21:49:48

%S 1,1,4,22,139,958,6979,52851,411884,3281684,26609931,218874331,

%T 1821767351,15315464340,129859965329,1109239893974,9536166375605,

%U 82449167265098,716449009997437,6253709697731562,54808237437608982,482103739329417219

%N G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n^2) * A(x)^(3*n+1).

%F The g.f. A(x) satisfies:

%F (1) 1 = Sum_{n>=0} (-x)^n*A(x)^(3*n+1) * Product_{k=1..n} (1 + x^(4*k-3)*A(x)^3)/(1 + x^(4*k-1)*A(x)^3);

%F (2) 1 = A(x)/(1+ x*A(x)^3/(1- x*(1-x^2)*A(x)^3/(1+ x^5*A(x)^3/(1- x^3*(1-x^4)*A(x)^3/(1+ x^9*A(x)^3/(1- x^5*(1-x^6)*A(x)^3/(1+ x^13*A(x)^3/(1- x^7*(1-x^8)*A(x)^3/(1- ...))))))))) (continued fraction);

%F due to identities of a partial elliptic theta function.

%e G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 139*x^4 + 958*x^5 + 6979*x^6 +...

%e which satisfies:

%e 1 = A(x) - x*A(x)^4 + x^4*A(x)^7 - x^9*A(x)^10 + x^16*A(x)^13 -+...

%e Related expansions.

%e A(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 965*x^4 + 7028*x^5 +...

%e A(x)^7 = 1 + 7*x + 49*x^2 + 357*x^3 + 2688*x^4 + 20811*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)^(3*m+1) ), #A-1)); if(n<0, 0, A[n+1])}

%Y Cf. A193111, A193112, A193113, A193114, A193115.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 16 2011