A357223 Coefficients in the power series A(x) such that: x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

1, 1, 4, 25, 164, 1177, 8887, 69748, 563232, 4649672, 39063521, 332904462, 2870862974, 25005954906, 219675658337, 1944131038267, 17316793719372, 155122164103293, 1396584226654493, 12630315100857638, 114687815080027358, 1045218902425525155, 9557367319452886097

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

G.f. A(x) satisfies:

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

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

(3) x*A(x)^3 = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.

(4) -x*A(x)^4 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.

Example

G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 164*x^4 + 1177*x^5 + 8887*x^6 + 69748*x^7 + 563232*x^8 + 4649672*x^9 + 39063521*x^10 + ...
such that
x*A(x)^3 = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...

Prog

(PARI) {a(n, p=3) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n)), ceil(sqrt(n)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A355357, A357221, A357222, A357224, A357225, A357226.

Sequence in context: A208264 A357162 A184755 * A074422 A128419 A372216

Adjacent sequences: A357220 A357221 A357222 * A357224 A357225 A357226

Keyword

nonn

Author

Paul D. Hanna, Sep 18 2022

Status

approved