A357208 Coefficients in the power series A(x) such that: x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

1, 1, 8, 74, 758, 8412, 98605, 1201739, 15075377, 193374064, 2524704727, 33440460233, 448246477551, 6069174992443, 82884604316537, 1140361539606239, 15791577929661603, 219930850717175458, 3078540089119391233, 43287917046150591163, 611156850554916771425

Offset: 0

Author

Paul D. Hanna, Sep 18 2022

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 74*x^3 + 758*x^4 + 8412*x^5 + 98605*x^6 + 1201739*x^7 + 15075377*x^8 + 193374064*x^9 + 2524704727*x^10 + ...
where
x*A(x)^4 = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...
		

Crossrefs

Cf. A355361, A357206, A357207, A357209.

Programs

PARI

{a(n) = my(A=[1, 1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( x*Ser(A)^4 - sum(m=-t, t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following relations.

(1) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

(2) -x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.

(3) x*A(x)^4 = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n), due to the Jacobi triple product identity.

(4) -x*A(x)^5 = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n), due to the Jacobi triple product identity.