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%I #10 Mar 12 2021 22:24:46
%S 1,4,4,-1,0,4,1,0,0,1,0,-8,-1,0,-8,0,0,4,1,0,16,-2,0,16,0,0,-4,2,0,
%T -32,-3,0,-32,1,0,8,4,0,56,-4,0,56,1,0,-16,4,0,-96,-6,0,-92,1,0,24,5,
%U 0,160,-8,0,152,1,0,-40,8,0,-252,-10,0,-240,2,0,64,11,0,392,-14,0,368,4,0,-96,14,0,-600,-19,0,-560,4,0
%N Expansion of q^(-1) * phi^2(q) * chi^3(q^9) / (chi(q^3) * phi^2(q^9)) in powers of q where phi(), 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="/A164612/b164612.txt">Table of n, a(n) for n = -1..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 chi(q^3)^3 / (q * chi(q)) + 4 + 4 * q * chi(q) / chi(q^3)^3 in powers of q where chi() is a Ramanujan theta function.
%F Expansion of eta(q^2)^10 * eta(q^3) * eta(q^9) * eta(q^12) * eta(q^36) / (eta(q)^4 * eta(q^4)^4 * eta(q^6)^2 * eta(q^18)^4) in powers of q.
%F Euler transform of period 36 sequence [ 4, -6, 3, -2, 4, -5, 4, -2, 2, -6, 4, -2, 4, -6, 3, -2, 4, -2, 4, -2, 3, -6, 4, -2, 4, -6, 2, -2, 4, -5, 4, -2, 3, -6, 4, 0, ...].
%F a(3*n) = 0 unless n=0. a(n) = A164268(n) unless n=0.
%F Convolution of A164613 and A062244.
%e G.f. = 1/q + 4 + 4*q - q^2 + 4*q^4 + q^5 + q^8 - 8*q^10 - q^11 - 8*q^13 + ...
%t eta[x_] := QPochhammer[x]; A164612[n_] := SeriesCoefficient[eta[q^2]^10* eta[q^3]*eta[q^9]*eta[q^12]*eta[q^36]/(eta[q]^4*eta[q^4]^4*eta[q^6]^2 *eta[q^18]^4), {q, 0, n}]; Table[A164612[n], {n, 0, 50}] (* _G. C. Greubel_, Aug 10 2017 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^3 + A) * eta(x^9 + A) * eta(x^12 + A) * eta(x^36 + A) / (eta(x + A)^4 * eta(x^4 + A)^4 * eta(x^6 + A)^2 * eta(x^18 + A)^4), n))};
%Y Cf. A062244, A164268, A164613.
%K sign
%O -1,2
%A _Michael Somos_, Aug 17 2009
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