A193116 G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n^2) * A(x)^(3*n+1).

1, 1, 4, 22, 139, 958, 6979, 52851, 411884, 3281684, 26609931, 218874331, 1821767351, 15315464340, 129859965329, 1109239893974, 9536166375605, 82449167265098, 716449009997437, 6253709697731562, 54808237437608982, 482103739329417219

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

The g.f. A(x) satisfies:

(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);

(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);

due to identities of a partial elliptic theta function.

Example

G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 139*x^4 + 958*x^5 + 6979*x^6 +...
which satisfies:
1 = A(x) - x*A(x)^4 + x^4*A(x)^7 - x^9*A(x)^10 + x^16*A(x)^13 -+...
Related expansions.
A(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 965*x^4 + 7028*x^5 +...
A(x)^7 = 1 + 7*x + 49*x^2 + 357*x^3 + 2688*x^4 + 20811*x^5 +...

Prog

(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])}

Crossrefs

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

Sequence in context: A025756 A366119 A200731 * A187254 A325453 A216712

Adjacent sequences: A193113 A193114 A193115 * A193117 A193118 A193119

Keyword

nonn

Author

Paul D. Hanna, Jul 16 2011

Status

approved