A213618 Expansion of phi(-q^3) * b(q^8) in powers of q where phi() is a Ramanujan theta function and b() is a cubic AGM theta function.
1, 0, 0, -2, 0, 0, 0, 0, -3, 0, 0, 6, 2, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 6, 0, 0, -14, 0, 0, 0, 0, -3, 0, 0, 12, 12, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 2, 0, 0, -12, 0, 0, 0, 0, -12, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, -12, 0, 0, 0, 18, 0, 0, -14, 0, 0, 0, 0
Offset: 0
Keywords
Examples
G.f. = 1 - 2*q^3 - 3*q^8 + 6*q^11 + 2*q^12 - 6*q^20 + 6*q^24 - 14*q^27 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..2500
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3] QPochhammer[ q^8]^3 / QPochhammer[ q^24], {q, 0, n}]; (* Michael Somos, Aug 26 2015 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^8 + A)^3 / (eta(x^6 + A) * eta(x^24 + A)), n))};
Formula
Expansion of eta(q^3)^2 * eta(q^8)^3 / (eta(q^6) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 0, 0, -2, 0, 0, -1, 0, -3, -2, 0, 0, -1, 0, 0, -2, -3, 0, -1, 0, 0, -2, 0, 0, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 93312^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A213607.
a(3*n + 1) = a(4*n + 1) = a(4*n + 2) = a(24*n + 15) = a(24*n + 23) = 0.
Comments