A257653 Expansion of f(-x^2)^3 * phi(x^3) / f(-x^6) in powers of x where phi(), f() are Ramanujan theta functions.
1, 0, -3, 2, 0, -6, 6, 0, -3, 12, 0, -6, 2, 0, -12, 0, 0, -12, 18, 0, -6, 12, 0, 0, 6, 0, -18, 14, 0, -18, 12, 0, -3, 12, 0, -12, 12, 0, -18, 0, 0, -24, 12, 0, -6, 36, 0, 0, 2, 0, -21, 12, 0, -18, 42, 0, -12, 12, 0, -18, 0, 0, -24, 0, 0, -24, 24, 0, -12, 24, 0
Offset: 0
Keywords
Examples
G.f. = 1 - 3*x^2 + 2*x^3 - 6*x^5 + 6*x^6 - 3*x^8 + 12*x^9 - 6*x^11 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Magma
A := Basis( ModularForms( Gamma0(36), 3/2), 71); A[1] - 3*A[3] + 2*A[4] + 6*A[6];
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Mathematica
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 EllipticTheta[ 3, 0, x^3] / QPochhammer[ x^6], {x, 0, n}]; -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^6 + A)^4 / (eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};
Formula
Expansion of eta(q^2)^3 * eta(q^6)^4 / (eta(q^3)^2 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ 0, -3, 2, -3, 0, -5, 0, -3, 2, -3, 0, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A257651.
G.f.: Product_{k>0} (1 - x^(2*k))^3 * (1 + x^(3*k))^2 / (1 + x^(6*k))^2.
Comments