A213607 Expansion of psi(x^4) * f(-x^3)^3 / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.
1, 1, 2, 0, 3, 2, 4, 0, 3, 3, 4, 0, 4, 3, 6, 0, 7, 3, 4, 0, 6, 5, 4, 0, 7, 4, 8, 0, 7, 5, 8, 0, 5, 4, 10, 0, 8, 5, 6, 0, 7, 7, 8, 0, 11, 5, 10, 0, 9, 8, 8, 0, 5, 4, 12, 0, 14, 5, 8, 0, 10, 8, 8, 0, 11, 8, 10, 0, 10, 9, 14, 0, 10, 5, 10, 0, 15, 7, 6, 0, 10, 9
Offset: 0
Keywords
Examples
1 + x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^6 + 3*x^8 + 3*x^9 + 4*x^10 + ... q^5 + q^11 + 2*q^17 + 3*q^29 + 2*q^35 + 4*q^41 + 3*q^53 + 3*q^59 + 4*q^65 + ...
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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Mathematica
QP := QPochhammer; a[n_]:= SeriesCoefficient[(QP[q^3]^3*QP[q^8]^2 )/( QP[q]*QP[q^4]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 07 2018 *) -
PARI
{a(n) = local(A); if ( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^8 + A)^2 / (eta(x + A) * eta(x^4 + A)), n))}
Formula
Expansion of q^(-5/6) * eta(q^3)^3 * eta(q^8)^2 / (eta(q) * eta(q^4)) in powers of q.
Euler transform of period 24 sequence [ 1, 1, -2, 2, 1, -2, 1, 0, -2, 1, 1, -1, 1, 1, -2, 0, 1, -2, 1, 2, -2, 1, 1, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 32^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A213618.
a(4*n + 3) = 0. a(4*n + 2) = 2 * A213023(n).
Comments