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%I #13 Mar 12 2021 22:24:45
%S 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,
%T -9,-16,28,0,0,-14,-25,44,0,0,-22,-38,67,0,0,-32,-57,100,0,0,-48,-84,
%U 146,0,0,-70,-121,210,0,0,-99,-172,299,0,0,-140,-243,420,0
%N Expansion of f(-x^4) * chi(x^5) / f(-x^5) in powers of x where f(), chi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A146164/b146164.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(1/4) * eta(q^4) * eta(q^10)^2 / (eta(q^5)^2 * eta(q^20)) in powers of q.
%F 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, ...].
%F 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.
%F a(5*n + 1) = a(5*n + 2) = 0.
%F a(n) = A138532(2*n + 1). a(5*n + 4) = - A146163(n).
%F Convolution inverse of A146165.
%e 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 + ...
%e G.f. = 1/q - q^15 + 2*q^19 - q^31 - 2*q^35 + 3*q^39 - 2*q^51 - 3*q^55 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^4] QPochhammer[ -x^5, x^10] / QPochhammer[ x^5], {x, 0, n}]; (* _Michael Somos_, Sep 03 2015 *)
%o (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))};
%Y Cf. A138532, A146162, A146163, A146165.
%K sign
%O 0,6
%A _Michael Somos_, Oct 27 2008, Nov 10 2008
%E Edited by _N. J. A. Sloane_, Nov 21 2008 at the suggestion of _R. J. Mathar_
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