A259538 Expansion of psi(-x)^2 / psi(-x^3) in powers of x where psi() is a Ramanujan theta function.
1, -2, 1, -1, 0, 1, 2, -2, 1, 1, -2, 0, 1, -4, 1, 1, -2, 2, 3, -2, 2, 1, -4, 2, 2, -6, 3, 2, -4, 3, 2, -6, 3, 4, -8, 3, 5, -10, 5, 3, -10, 6, 7, -10, 5, 8, -12, 6, 7, -16, 8, 7, -16, 9, 9, -18, 10, 10, -22, 10, 11, -26, 13, 12, -26, 15, 14, -28, 15, 17, -34
Offset: 0
Keywords
Examples
G.f. = 1 - 2*x + x^2 - x^3 + x^5 + 2*x^6 - 2*x^7 + x^8 + x^9 - 2*x^10 + ... G.f. = 1/q - 2*q^7 + q^15 - q^23 + q^39 + 2*q^47 - 2*q^55 + q^63 + ...
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
Crossrefs
Cf. A259529.
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 2, 0, 1, 2, 2, 0, 2, 2, 1, 0, 2, 1}[[Mod[k, 12, 1]]], {k, n}], {x, 0, n}]; a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] QPochhammer[ x^4])^2 / ( QPochhammer[ x^3, x^6] QPochhammer[ x^12]), {x, 0, n}]; -
PARI
{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, 2, 0, 1, 2, 2, 0, 2, 2, 1, 0, 2][k%12 + 1]), n))}; -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A) / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A)), n))};
Formula
Expansion of q^(1/8) * eta(q)^2 * eta(q^4)^2 * eta(q^6) / (eta(q^2)^2 * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -2, 0, -1, -2, -2, 0, -2, -2, -1, 0, -2, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (768 t)) = 12 (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A259529.
G.f.: Product_{k>0} (1 - x^k) * (1 + x^(2*k)) / ((1 + x^k + x^(2*k)) * (1 - x^(2*k) + x^(4*k))).
Comments