A357224 Coefficients in the power series A(x) such that: x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

1, 1, 5, 38, 315, 2855, 27325, 272030, 2788042, 29221793, 311767823, 3374650902, 36968040004, 409076635878, 4565873250981, 51342245169913, 581093383193700, 6614534942714496, 75675364150733073, 869713202188274489, 10036085000519702155, 116238137830534589525

Offset

0,3

Links

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

Formula

G.f. A(x) satisfies:

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

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

(3) 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.

(4) -x*A(x)^5 = 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 + 5*x^2 + 38*x^3 + 315*x^4 + 2855*x^5 + 27325*x^6 + 272030*x^7 + 2788042*x^8 + 29221793*x^9 + 311767823*x^10 + ...
such that
x*A(x)^4 = ... + 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=4) = 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, A357223, A357225, A357226.

Sequence in context: A073508 A282964 A357163 * A247773 A207411 A316598

Adjacent sequences: A357221 A357222 A357223 * A357225 A357226 A357227

Keyword

nonn

Author

Paul D. Hanna, Sep 18 2022

Status

approved