A146164 Expansion of f(-x^4) * chi(x^5) / f(-x^5) in powers of x where f(), chi() are Ramanujan theta functions.
1, 0, 0, 0, -1, 2, 0, 0, -1, -2, 3, 0, 0, -2, -3, 6, 0, 0, -3, -6, 11, 0, 0, -6, -10, 18, 0, 0, -9, -16, 28, 0, 0, -14, -25, 44, 0, 0, -22, -38, 67, 0, 0, -32, -57, 100, 0, 0, -48, -84, 146, 0, 0, -70, -121, 210, 0, 0, -99, -172, 299, 0, 0, -140, -243, 420, 0
Offset: 0
Keywords
Examples
G.f. = 1 - x^4 + 2*x^5 - x^8 - 2*x^9 + 3*x^10 - 2*x^13 - 3*x^14 + 6*x^15 + ... G.f. = 1/q - q^15 + 2*q^19 - q^31 - 2*q^35 + 3*q^39 - 2*q^51 - 3*q^55 + ...
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
a[ n_] := SeriesCoefficient[ QPochhammer[ x^4] QPochhammer[ -x^5, x^10] / QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Sep 03 2015 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^10 + A)^2 / (eta(x^5 + A)^2 * eta(x^20 + A)), n))};
Formula
Expansion of q^(1/4) * eta(q^4) * eta(q^10)^2 / (eta(q^5)^2 * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ 0, 0, 0, -1, 2, 0, 0, -1, 0, 0, 0, -1, 0, 0, 2, -1, 0, 0, 0, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = (5/4)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A146162.
a(5*n + 1) = a(5*n + 2) = 0.
Convolution inverse of A146165.
Extensions
Edited by N. J. A. Sloane, Nov 21 2008 at the suggestion of R. J. Mathar
Comments