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A146164
Expansion of f(-x^4) * chi(x^5) / f(-x^5) in powers of x where f(), chi() are Ramanujan theta functions.
4
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
0,6
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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.
a(n) = A138532(2*n + 1). a(5*n + 4) = - A146163(n).
Convolution inverse of A146165.
EXAMPLE
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 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^4] QPochhammer[ -x^5, x^10] / QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Sep 03 2015 *)
PROG
(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))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 27 2008, Nov 10 2008
EXTENSIONS
Edited by N. J. A. Sloane, Nov 21 2008 at the suggestion of R. J. Mathar
STATUS
approved