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A134078
Expansion of (phi(-q) / phi(-q^2))^3 * phi(q^3)^5 / phi(-q^6) in powers of q where phi() is a Ramanujan theta function.
4
1, -6, 18, -34, 42, -36, 30, -48, 90, -118, 108, -72, 54, -84, 144, -204, 186, -108, 66, -120, 252, -272, 216, -144, 102, -186, 252, -370, 336, -180, 180, -192, 378, -408, 324, -288, 90, -228, 360, -476, 540, -252, 240, -264, 504, -708, 432, -288, 198, -342
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
Euler transform of period 12 sequence [ -6, 3, 4, 0, -6, -10, -6, 0, 4, 3, -6, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 8 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133739.
a(3*n + 2) = 18 * A134079(n). a(6*n + 5) = -36 * A098098(n).
EXAMPLE
G.f. = 1 - 6*x + 18*x^2 - 34*x^3 + 42*x^4 - 36*x^5 + 30*x^6 - 48*x^7 + 90*x^8 + ...
MATHEMATICA
a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, -q]/EllipticTheta[3, 0, -q^2])^3*(EllipticTheta[3, 0, q^3]^5/EllipticTheta[3, 0, -q^6]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 22 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^6 * eta(x^4 + A)^3 * eta(x^6 + A)^23 / ( eta(x^2 + A)^9 * eta(x^3 + A)^10 * eta(x^12 + A)^9 ), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 06 2007
STATUS
approved