A357207 Coefficients in the power series A(x) such that: x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

1, 1, 7, 55, 469, 4307, 41678, 418872, 4330275, 45754091, 491916135, 5364166402, 59186372395, 659556170091, 7412556531714, 83921355689635, 956228695216241, 10957322339242547, 126189988012692329, 1459793848341094130, 16955390069787782159, 197653935181097885580

Offset

0,3

Links

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

Formula

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

(1) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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

(3) x*A(x)^3 = 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)^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.

Example

G.f.: A(x) = 1 + x + 7*x^2 + 55*x^3 + 469*x^4 + 4307*x^5 + 41678*x^6 + 418872*x^7 + 4330275*x^8 + 45754091*x^9 + 491916135*x^10 + ...
where
x*A(x)^3 = ... - 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 + ...

Prog

(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)^3 - 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), ", "))

Crossrefs

Cf. A355361, A357206, A357208, A357209.

Sequence in context: A246459 A152262 A078018 * A108628 A116862 A096307

Adjacent sequences: A357204 A357205 A357206 * A357208 A357209 A357210

Keyword

nonn

Author

Paul D. Hanna, Sep 18 2022

Status

approved