OFFSET
1,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..2500
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^7) * eta(q^21) / (eta(q) * eta(q^3)) in powers of q.
Euler transform of period 21 sequence [1, 1, 2, 1, 1, 2, 0, 1, 2, 1, 1, 2, 1, 0, 2, 1, 1, 2, 1, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (1 + 7 * u*v) - (u+v) * (u^2 - 3 * u*v + v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (21 t)) = 1/7 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226006.
G.f.: x * Product_{k>0} (1 - x^(7*k)) * (1 - x^(21*k)) / ( (1 - x^k) * (1 - x^(3*k))).
Convolution inverse of A226006.
a(n) ~ exp(4*Pi*sqrt(n/21)) / (sqrt(2) * 3^(1/4) * 7^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
EXAMPLE
G.f. = q + q^2 + 2*q^3 + 4*q^4 + 6*q^5 + 9*q^6 + 16*q^7 + 21*q^8 + 32*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^7] QPochhammer[ q^21] / (QPochhammer[ q] QPochhammer[ q^3]), {q, 0, n}]; (* Michael Somos, Apr 12 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^7 + A) * eta(x^21 + A) / (eta(x + A) * eta(x^3 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, May 22 2013
STATUS
approved