login
A242609
Expansion of phi(-q) * phi(q^8) in powers of q where phi() is a Ramanujan theta function.
3
1, -2, 0, 0, 2, 0, 0, 0, 2, -6, 0, 0, 4, 0, 0, 0, 2, -4, 0, 0, 0, 0, 0, 0, 4, -2, 0, 0, 0, 0, 0, 0, 2, -8, 0, 0, 6, 0, 0, 0, 0, -4, 0, 0, 4, 0, 0, 0, 4, -2, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 6, -4, 0, 0, 4, 0, 0, 0, 0, -10, 0
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q)^2 * eta(q^16)^5 / (eta(q^2) * eta(q^8)^2 * eta(q^32)^2) in powers of q.
G.f.: (Sum_{k in Z} (-x)^k^2) * (Sum_{k in Z} (x^8)^k^2).
a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = 0. a(4*n) = a(8*n) = A033715(n). a(8*n + 1) = -2 * A112603(n). a(8*n + 4) = 2 * A113411(n).
a(n) = (-1)^n * A226225(n).
EXAMPLE
G.f. = 1 - 2*q + 2*q^4 + 2*q^8 - 6*q^9 + 4*q^12 + 2*q^16 - 4*q^17 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^8], {q, 0, n}];
PROG
(PARI) {a(n) = if( n<1, n==0, 2 * (-1)^n * (n%4 < 2) * sumdiv( n, d, kronecker( -2, d)))};
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^16 + A)^5 / (eta(x^2 + A) * eta(x^8 + A)^2 * eta(x^32 + A)^2), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, May 19 2014
STATUS
approved