OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^2)^3 * eta(q^9)^2 * eta(q^36) / (eta(q)^2 * eta(q^4) * eta(q^18)^3) in powers of q.
G.f. A(x) = B(x) / B(x^9) where B(x) is the g.f. of A080054.
Euler transform of period 36 sequence [ 2, -1, 2, 0, 2, -1, 2, 0, 0, -1, 2, 0, 2, -1, 2, 0, 2, 0, 2, 0, 2, -1, 2, 0, 2, -1, 0, 0, 2, -1, 2, 0, 2, -1, 2, 0, ...].
a(n) = (-1)^n * A260215(n). - Michael Somos, Aug 14 2015
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
EXAMPLE
G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 22*x^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2] QPochhammer[ -q, q] QPochhammer[ q^9, q^18] QPochhammer[ q^9, -q^9], {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^9 + A)^2 * eta(x^36 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^18 + A)^3), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 10 2015
STATUS
approved