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
Euler transform of period 12 sequence [ 8, -12, 6, -4, 8, -9, 8, -4, 6, -12, 8, -3, ...].
a(n) = A004013(12*n).
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = Pi^(5/4) * 3^(1/4) * Gamma(11/12) * (3^(1/2)-1) / Gamma(2/3) / Gamma(3/4)^4 = A388929. - Simon Plouffe, Sep 21 2025
EXAMPLE
G.f. = 1 + 8*x + 24*x^2 + 30*x^3 + 8*x^4 + 36*x^6 + 48*x^7 + 24*x^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^4 / EllipticTheta[ 3, 0, q^3], {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^20 * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^8 * eta(x^4 + A)^8 * eta(x^6 + A)^5), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 17 2015
STATUS
approved
