A185653 Expansion of exp( Sum_{n>=1} -3*sigma(2n)*x^n/n ) in powers of x.

1, -9, 30, -39, 0, 18, 49, 0, -192, 110, 81, 78, -130, 0, -30, -121, 0, 210, 320, -270, 0, -407, 0, 192, 190, 0, 0, 0, 351, -210, -418, 0, -510, 448, 0, 462, 611, 0, -960, 50, 0, 0, -350, 0, 450, -361, -162, 960, 0, 0, 798, -782, 0, -1170, -290, -441, 702, 850, 0, 0, 576

Offset

0,2

Comments

When is a(n) zero (A258867)?

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1024

Formula

Expansion of q^(-1/8)*eta(q)^9/eta(q^2)^3 in powers of q; equals the self-convolution cube of A115110 [See formula of Michael Somos for A115110].

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/4) * exp(Pi / 8) * Pi^(3/2) * 2^(1/8) / Gamma(3/4)^6 = A388696. - Simon Plouffe, Sep 18 2025

Example

G.f. = 1 - 9*x + 30*x^2 - 39*x^3 + 18*x^5 + 49*x^6 - 192*x^8 + 110*x^9 + ...

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, -3*sigma(2*m)*x^m/m)+x*O(x^n)), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(eta(X)^9/eta(X^2)^3, n)}

Crossrefs

Cf. A115110, A000203, A258867.

Sequence in context: A279618 A158503 A179506 * A326150 A167154 A063150

Adjacent sequences: A185650 A185651 A185652 * A185654 A185655 A185656

Keyword

sign

Author

Paul D. Hanna, Feb 16 2011

Status

approved