A357206 Coefficients in the power series A(x) such that: x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

1, 1, 6, 39, 267, 1949, 14927, 118517, 966840, 8055107, 68247637, 586231174, 5093508706, 44685394843, 395287384067, 3521909281230, 31576985230764, 284687856687607, 2579319718212675, 23472206080648463, 214448766193151410, 1966300700448875377, 18088031500652556354

Offset

0,3

Links

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

Formula

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

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

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

(3) x*A(x)^2 = 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)^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.

Example

G.f.: A(x) = 1 + x + 6*x^2 + 39*x^3 + 267*x^4 + 1949*x^5 + 14927*x^6 + 118517*x^7 + 966840*x^8 + 8055107*x^9 + 68247637*x^10 + ...
where
x*A(x)^2 = ... - 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)^2 - 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, A357207, A357208, A357209.

Sequence in context: A006256 A052392 A370376 * A370026 A199491 A147961

Adjacent sequences: A357203 A357204 A357205 * A357207 A357208 A357209

Keyword

nonn

Author

Paul D. Hanna, Sep 18 2022

Status

approved