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%I #16 Feb 04 2021 06:04:42
%S 1,-1,-1,0,1,0,-1,1,2,0,-3,0,2,0,-3,0,5,0,-4,-2,4,0,-5,0,7,2,-7,0,5,0,
%T -10,1,12,0,-10,0,14,-4,-17,0,21,0,-22,4,24,0,-34,0,33,1,-36,0,45,0,
%U -45,-8,52,0,-55,0,62,8,-71,0,70,0,-88,2,96,0,-98,0
%N Expansion of eta(q) * eta(q^36) / (eta(q^4) * eta(q^9)) in powers of q.
%H G. C. Greubel, <a href="/A182038/b182038.txt">Table of n, a(n) for n = 1..2500</a>
%F Euler transform of period 36 sequence [-1, -1, -1, 0, -1, -1, -1, 0, 0, -1, -1, 0, -1, -1, -1, 0, -1, 0, -1, 0, -1, -1, -1, 0, -1, -1, 0, 0, -1, -1, -1, 0, -1, -1, -1, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = f(t) where q = exp(2 Pi i t).
%F G.f.: Product_{k>0} (1 - x^k) * (1 - x^(36*k)) / ((1 - x^(4*k)) * (1 - x^9*k)).
%F a(6*n) = a(6*n + 4) = 0. a(6*n + 2) = -A092848(n).
%F Convolution inverse of A187020.
%e G.f. = q - q^2 - q^3 + q^5 - q^7 + q^8 + 2*q^9 - 3*q^11 + 2*q^13 - 3*q^15 + ...
%t QP = QPochhammer; s = QP[q]*(QP[q^36]/(QP[q^4]*QP[q^9])) + O[q]^80; CoefficientList[s, q]
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^36 + A) / (eta(x^4 + A) * eta(x^9 + A)), n))};
%Y Cf. A092848, A187020. Essentially the same as A213266.
%K sign
%O 1,9
%A _Michael Somos_, Apr 07 2012