A357225 Coefficients in the power series A(x) such that: x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

1, 1, 6, 54, 542, 5950, 69089, 834807, 10387628, 132206325, 1713016233, 22520857313, 299667203315, 4028078782339, 54615552455056, 746073353306341, 10258385111897258, 141862903772876529, 1971827463536643265, 27532294076219156008, 386001188585539328720

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

G.f. A(x) satisfies:

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

(2) -x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.

(3) 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.

(4) -x*A(x)^6 = 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 + 6*x^2 + 54*x^3 + 542*x^4 + 5950*x^5 + 69089*x^6 + 834807*x^7 + 10387628*x^8 + 132206325*x^9 + 1713016233*x^10 + ...
such that
x*A(x)^5 = ... + 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=5) = 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, A357224, A357226.

Sequence in context: A228413 A098658 A357164 * A109576 A365843 A366014

Adjacent sequences: A357222 A357223 A357224 * A357226 A357227 A357228

Keyword

nonn

Author

Paul D. Hanna, Sep 18 2022

Status

approved