A357226 Coefficients in the power series A(x) such that: x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

1, 1, 7, 73, 861, 11112, 151822, 2159143, 31627140, 473909468, 7230035454, 111924733904, 1753728878625, 27759947012294, 443247756591472, 7130680715081049, 115466397372003479, 1880525144522628300, 30783524695736369568, 506215648672559259036, 8358521379108937920413

Offset

0,3

Links

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

Formula

G.f. A(x) satisfies:

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

(2) -x*A(x)^7 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.

(3) 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.

(4) -x*A(x)^7 = 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 + 7*x^2 + 73*x^3 + 861*x^4 + 11112*x^5 + 151822*x^6 + 2159143*x^7 + 31627140*x^8 + 473909468*x^9 + 7230035454*x^10 + ...
such that
x*A(x)^6 = ... + 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=6) = 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, A357225.

Sequence in context: A099141 A084768 A357165 * A106651 A114429 A365846

Adjacent sequences: A357223 A357224 A357225 * A357227 A357228 A357229

Keyword

nonn

Author

Paul D. Hanna, Sep 18 2022

Status

approved