OFFSET
-1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1) * phi(q) / psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2)^5 * eta(q^8) / (eta(q)^2 * eta(q^4)^2 * eta(q^16)^2) in powers of q.
Euler transform of period 16 sequence [ 2, -3, 2, -1, 2, -3, 2, -2, 2, -3, 2, -1, 2, -3, 2, 0, ...].
G.f. A(x) satisfies: 0 = f(A(x), A(x^2)) where f(u, v) = v^2 - (v - 2) * (u^2 - 4*u + 8).
G.f.: 2 + (1/q) * Product_{k>0} ((1 + q^(8*k - 4)) / (1 + q^(8*k)))^2.
a(4*n - 1) = A029839(n). a(4*n) = 0 unless n=0. a(4*n + 1) = a(4*n + 2) = 0. Convolution inverse of A208605.
a(n) = -(-1)^n * A185338(n).
EXAMPLE
G.f. = 1/q + 2 + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 + 5*q^31 + ...
MATHEMATICA
QP = QPochhammer; s = QP[q^2]^5*(QP[q^8]/(QP[q]^2*QP[q^4]^2*QP[q^16]^2)) + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A) / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^16 + A)^2), n))}
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Feb 29 2012
STATUS
approved