OFFSET
1,2
REFERENCES
T. Gannon, Moonshine Beyond the Monster, Cambridge, 2006; see p. 433.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..720
FORMULA
a(n) ~ -(-1)^n * Gamma(3/4)^16 * exp(Pi*n) * n^3 / (24*Pi^4). - Vaclav Kotesovec, Jun 03 2018
Expansion of A/(q*(1+64*A)^2), with A = (eta(q^2)/eta(q))^24, in powers of q. - G. C. Greubel, Jun 19 2018
Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/648, validated to 900 digits. - Simon Plouffe, May 06 2023
EXAMPLE
G.f. = q - 104*q^2 + 6444*q^3 - 311744*q^4 + 13018830*q^5 - 493025760*q^6 + ...
MATHEMATICA
nmax = 20; CoefficientList[Series[1/(128*x + Product[1/(1 + x^k)^24, {k, 1, nmax}] + 4096*x^2*Product[(1 + x^k)^24, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 03 2018 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; x2:= (eta[q^2]/eta[q])^24; a:= CoefficientList[Series[x2/(1 + 64*x2)^2/q, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 19 2018 *)
PROG
(PARI) q='q+O('q^50); A = q*(eta(q^2)/eta(q))^24; Vec(A/(q*(1+64*A)^2)) \\ G. C. Greubel, Jun 19 2018
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, May 07 2017
STATUS
approved