A320049 Expansion of (psi(x) / phi(x))^6 in powers of x where phi(), psi() are Ramanujan theta functions.

1, -6, 27, -98, 309, -882, 2330, -5784, 13644, -30826, 67107, -141444, 289746, -578646, 1129527, -2159774, 4052721, -7474806, 13569463, -24274716, 42838245, -74644794, 128533884, -218881098, 368859591, -615513678, 1017596115, -1667593666, 2710062756, -4369417452

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Simon Plouffe, Numbers in the base e^Pi, arXiv:2509.15609 [math.NT], 2025. See p. 18/24, marked 211.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Convolution inverse of A029843.

Expansion of q^(-3/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^6 in powers of q.

a(n) ~ (-1)^n * 3^(1/4) * exp(Pi*sqrt(3*n)) / (128*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Oct 06 2018

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/16) * exp(3*Pi/4) * 2^(1/4) = A389043. - Simon Plouffe, Sep 22 2025

Mathematica

nmax = 40; CoefficientList[Series[Product[((1-x^k) * (1-x^(4*k))^2 / (1-x^(2*k))^3)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)

Crossrefs

(psi(x) / phi(x))^b: A083365 (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), A195861 (b=5), this sequence (b=6), A320050 (b=7).

Cf. A029843.

Sequence in context: A136747 A278357 A001940 * A121591 A071734 A160507

Adjacent sequences: A320046 A320047 A320048 * A320050 A320051 A320052

Keyword

sign

Author

Seiichi Manyama, Oct 04 2018

Status

approved