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%I #13 Mar 12 2021 22:24:46
%S 1,2,0,0,4,4,0,0,2,-2,0,0,-8,-4,0,0,-1,6,0,0,20,4,0,0,-2,-8,0,0,-40,
%T -8,0,0,3,10,0,0,72,16,0,0,2,-16,0,0,-128,-20,0,0,-4,22,0,0,220,24,0,
%U 0,-4,-30,0,0,-360,-36,0,0,5,44,0,0,576,52,0,0,8
%N Expansion of q^-2 * phi(q) * phi(q^4) / psi(q^8)^2 in powers of q where phi(), psi() 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="/A185146/b185146.txt">Table of n, a(n) for n = -2..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">MathWorld: Ramanujan Theta Functions</a>
%F Expansion of eta(q^2)^5 * eta(q^8)^7 / (eta(q)^2 * eta(q^4)^4 * eta(q^16)^6) in powers of q.
%F Euler transform of period 16 sequence [ 2, -3, 2, 1, 2, -3, 2, -6, 2, -3, 2, 1, 2, -3, 2, 0, ...].
%F a(4*n) = a(4*n + 1) = 0. a(4*n - 2) = A029841(n). a(4*n - 1) = 2 * A029839(n).
%e q^-2 + 2*q^-1 + 4*q^2 + 4*q^3 + 2*q^6 - 2*q^7 - 8*q^10 - 4*q^11 - q^14 + ...
%t f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A185146[n_] := SeriesCoefficient[(f[x, x]*f[x^4, x^4])/(x*f[x^8, x^24])^2, {x, 0, n}]; Table[A185146[n], {n,-2,50}] (* _G. C. Greubel_, Jun 23 2017 *)
%o (PARI) {a(n) = local(A); if( n<-2, 0, n += 2; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^4 * eta(x^16 + A)^6), n))}
%Y Cf. A029839, A029841.
%K sign
%O -2,2
%A _Michael Somos_, Feb 28 2012
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