OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/2) * (eta(q) * eta(q^4) * eta(q^6)^3 / (eta(q^2)^3 * eta(q^3) * eta(q^12)))^6 in powers of q.
Euler transform of period 12 sequence [ -6, 12, 0, 6, -6, 0, -6, 6, 0, 12, -6, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = (1/27) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132107.
Convolution inverse of A132107.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(13/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018
EXAMPLE
G.f. = 1 - 6*x + 27*x^2 - 92*x^3 + 279*x^4 - 756*x^5 + 1913*x^6 - 4536*x^7 + ...
G.f. = q - 6*q^3 + 27*q^5 - 92*q^7 + 279*q^9 - 756*q^11 + 1913*q^13 - 4536*q^15 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3] / QPochhammer[ -x])^6, {x, 0, n}];
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/2)* (eta[q]*eta[q^4]*eta[q^6]^3/(eta[q^2]^3*eta[q^3]*eta[q^12]))^6, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 04 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3 / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A)))^6, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Nov 01 2015
STATUS
approved