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A258291
Expansion of q^(-1/4) * eta(q) * eta(q^2) * eta(q^6) / eta(q^3) in powers of q.
3
1, -1, -2, 2, -1, 0, 3, -1, 0, 2, -1, -4, 1, -1, 0, 2, -2, 0, 2, 0, -2, 4, -1, 0, 2, -1, 0, 2, -1, -4, 1, -2, 0, 0, -1, 0, 4, -1, -4, 2, 0, 0, 3, -1, 0, 2, -2, 0, 2, -1, 0, 4, 0, 0, 0, -2, -6, 2, -1, 0, 2, -1, 0, 0, -1, -4, 4, -1, 0, 2, -1, 0, 3, -1, 0, 0, -2
OFFSET
0,3
LINKS
FORMULA
Euler transform of period 6 sequence [ -1, -2, 0, -2, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 9 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258277.
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 + x^(3*k)).
a(3*n) = A002175(n). a(3*n + 1) = - A121444(n). a(9*n + 2) = -2 * A008441(n). a(9*n + 5) = a(9*n + 8) = 0.
EXAMPLE
G.f. = 1 - x - 2*x^2 + 2*x^3 - x^4 + 3*x^6 - x^7 + 2*x^9 - x^10 - 4*x^11 + ...
G.f. = q - q^5 - 2*q^9 + 2*q^13 - q^17 + 3*q^25 - q^29 + 2*q^37 - q^41 + ...
MATHEMATICA
QP := QPochhammer; CoefficientList[Series[QP[q]*QP[q^2]*QP[q^6]/QP[q^3], {q, 0, 50}], q]] (* G. C. Greubel, Aug 04 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^6 + A) / eta(x^3 + A), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, May 25 2015
STATUS
approved