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 (chi(-x) * psi(x^3))^2 in powers of x where chi(), psi() are Ramanujan theta functions.
Expansion of q^(-2/3) * (eta(q) * eta(q^6)^2 / (eta(q^2) * eta(q^3)))^2 in powers of q.
Euler transform of period 6 sequence [ -2, 0, 0, 0, -2, -2, ...].
a(2*n) = A122865(n). a(4*n +1) = -2 * A121444(n). a(4*n + 2) = A122856(n). a(4*n + 3) = a(8*n + 4) = 0. a(8*n) = A002175(n). a(8*n + 2) = A122856(n).
2 * a(n) = -A258210(3*n + 2). - Michael Somos, May 01 2016
EXAMPLE
G.f. = 1 - 2*x + x^2 - 2*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + x^10 - 2*x^13 + ...
G.f. = q^2 - 2*q^5 + q^8 - 2*q^17 + 2*q^20 + 2*q^26 - 2*q^29 + q^32 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 EllipticTheta[ 2, 0, x^(3/2)]^2 / (4 x^(3/4)), {x, 0, n}];
PROG
(PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A)))^2, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, May 25 2015
STATUS
approved