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%I #10 Mar 12 2021 22:24:46
%S 1,1,1,0,1,2,0,0,1,0,-2,-4,0,-2,-8,0,1,-2,0,4,14,0,4,24,0,-1,6,0,-8,
%T -38,0,-8,-63,0,2,-16,0,14,92,0,14,150,0,-4,36,0,-24,-208,0,-23,-329,
%U 0,6,-78,0,40,440,0,38,684,0,-10,160,0,-63,-884,0,-60
%N Expansion of c(q^2)^2 / (c(-q) * c(-q^3)) in powers of q where c() is a cubic AGM theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H G. C. Greubel, <a href="/A164615/b164615.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 (chi(q^3) * psi(-q^3)^2)^2 / (psi(-q) * f(q^9)^2) in powers of q where psi(), chi(), f() are Ramanujan theta functions.
%F Expansion of eta(q^2) * eta(q^3)^2 * eta(q^9)^3 * eta(q^12)^2 * eta(q^36)^3 / (eta(q) * eta(q^4) * eta(q^18)^9) in powers of q.
%F Euler transform of period 36 sequence [ 1, 0, -1, 1, 1, -2, 1, 1, -4, 0, 1, -3, 1, 0, -1, 1, 1, 4, 1, 1, -1, 0, 1, -3, 1, 0, -4, 1, 1, -2, 1, 1, -1, 0, 1, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258111. - _Michael Somos_, May 20 2015
%F a(3*n) = 0 unless n=0. a(3*n + 1) = A128111(n). a(3*n + 2) = A164614(n).
%F Convolution inverse of A164616.
%F a(n) = (-1)^n * A182034(n). - _Michael Somos_, May 20 2015
%e G.f. = 1 + q + q^2 + q^4 + 2*q^5 + q^8 - 2*q^10 - 4*q^11 - 2*q^13 - 8*q^14 + ...
%t eta[x_] := QPochhammer[x]; A164615[n_] := SeriesCoefficient[eta[q^2]* eta[q^3]^2*eta[q^9]^3*eta[q^12]^2*eta[q^36]^3/(eta[q]*eta[q^4] *eta[q^18]^9), {q, 0, n}]; Table[A164615[n], {n,0,50}] (* _G. C. Greubel_, Aug 10 2017 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^9 + A)^3 * eta(x^12 + A)^2 * eta(x^36 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^9), n))};
%Y Cf. A128111, A164614, A164616, A182034. A258111.
%K sign
%O 0,6
%A _Michael Somos_, Aug 17 2009
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