A320050 Expansion of (psi(x) / phi(x))^7 in powers of x where phi(), psi() are Ramanujan theta functions.

1, -7, 35, -140, 483, -1498, 4277, -11425, 28889, -69734, 161735, -362271, 786877, -1662927, 3428770, -6913760, 13660346, -26492361, 50504755, -94766875, 175221109, -319564227, 575387295, -1023624280, 1800577849, -3133695747, 5399228149, -9214458260, 15584195428

Offset

0,2

Comments

In general, for b > 0 and (psi(x) / phi(x))^b, a(n) ~ (-1)^n * b^(1/4) * exp(Pi*sqrt(b*(n/2))) / (2^(b + 7/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2018

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 A029844.

Expansion of q^(-7/8) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^7 in powers of q.

a(n) ~ (-1)^n * 7^(1/4) * exp(Pi*sqrt((7*n)/2)) / (256*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Oct 06 2018

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/32) * exp(7*Pi/8) * 2^(5/8) = A389044. - Simon Plouffe, Sep 22 2025

Mathematica

nmax = 50; CoefficientList[Series[Product[((1-x^k) * (1-x^(4*k))^2 / (1-x^(2*k))^3)^7, {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), A320049 (b=6), this sequence (b=7).

Cf. A029844.

Sequence in context: A327385 A344101 A001941 * A160460 A160539 A023006

Adjacent sequences: A320047 A320048 A320049 * A320051 A320052 A320053

Keyword

sign

Author

Seiichi Manyama, Oct 04 2018

Status

approved