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^4) * eta(q^6)^7 / (eta(q)^2 * eta(q^2) * eta(q^3)^2 * eta(q^12)^3) in powers of q.
Euler transform of period 12 sequence [ 2, 3, 4, 2, 2, -2, 2, 2, 4, 3, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/6) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187145.
a(n) = A261154(3*n).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Mar 17 2018
EXAMPLE
G.f. = 1 + 2*x + 6*x^2 + 14*x^3 + 30*x^4 + 60*x^5 + 114*x^6 + 208*x^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^6] QPochhammer[ -x^4, x^6] QPochhammer[ x^6] EllipticTheta[ 3, 0, x^3] / QPochhammer[ x]^2, {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x + A)^2 * eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^3), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Nov 08 2015
STATUS
approved