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
Expansion of f(-x) * psi(x^3)^3 / (f(x)^3 * psi(-x^3)) in powers of x where psi(), f() are Ramanujan theta functions.
Expansion of q^(-2/3) * eta(q)^4 * eta(q^4)^3 * eta(q^6)^7 / (eta(q^2)^9 * eta(q^3)^4 * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [-4, 5, 0, 2, -4, 2, -4, 2, 0, 5, -4, 0, ...].
2 * a(n) = A260215(3*n + 2).
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
EXAMPLE
G.f. = 1 - 4*x + 11*x^2 - 24*x^3 + 48*x^4 - 92*x^5 + 170*x^6 - 304*x^7 + ...
G.f. = q^2 - 4*q^5 + 11*q^8 - 24*q^11 + 48*q^14 - 92*q^17 + 170*q^20 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 2^(-5/2) x^(-3/4) QPochhammer[ x] / QPochhammer[ -x]^3 EllipticTheta[ 2, 0, x^(3/2)]^3 / EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A)^3 * eta(x^6 + A)^7 / (eta(x^2 + A)^9 * eta(x^3 + A)^4 * eta(x^12 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Nov 08 2015
STATUS
approved