OFFSET
1,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^4) * eta(q^6) * eta(q^9) * eta(q^36)^2 / (eta(q) * eta(q^12) * eta(q^18)^3) in powers of q.
Euler transform of period 36 sequence [ 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, ...].
a(2*n) = A123629(n).
a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
EXAMPLE
G.f. = q + q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 6*q^6 + 8*q^7 + 11*q^8 + 14*q^9 + ...
MATHEMATICA
nmax=60; CoefficientList[Series[Product[(1-x^(4*k)) * (1-x^(6*k)) * (1-x^(9*k)) * (1+x^(18*k))^2 / ((1-x^k) * (1-x^(12*k)) * (1-x^(18*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
QP := QPochhammer; A233693[n_]:= SeriesCoefficient[QP[q^4]*QP[q^6] *QP[q^9]*QP[q^36]^2/(QP[q]* QP[q^12]*QP[q^18]^3), {q, 0, n}]; Table[A233693[n], {n, 0, 50}] (* G. C. Greubel, Dec 25 2017 *)
PROG
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A) * eta(x^9 + A) * eta(x^36 + A)^2 / (eta(x + A) * eta(x^12 + A) * eta(x^18 + A)^3), n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Dec 14 2013
STATUS
approved