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A164616 Expansion of c(-q) * c(-q^3) / c(q^2)^2 in powers of q where c() is a cubic AGM theta function.

%I A164616 #11 Feb 16 2025 08:33:11
%S A164616 1,-1,0,1,-2,0,4,-5,0,10,-12,0,20,-26,0,39,-50,0,76,-92,0,140,-168,0,
%T A164616 244,-295,0,415,-496,0,696,-818,0,1140,-1332,0,1820,-2126,0,2861,
%U A164616 -3324,0,4448,-5126,0,6816,-7824,0,10292,-11793,0,15372,-17548,0,22756
%N A164616 Expansion of c(-q) * c(-q^3) / c(q^2)^2 in powers of q where c() is a cubic AGM theta function.
%C A164616 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A164616 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A164616 G. C. Greubel, <a href="/A164616/b164616.txt">Table of n, a(n) for n = 0..1000</a>
%H A164616 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A164616 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A164616 Expansion of (psi(-q) * f(q^9)^3) / (chi(q^3) * psi(-q^3)^2)^2 in powers of q where psi(), chi(), f() are Ramanujan theta functions.
%F A164616 Expansion of eta(q) * eta(q^4) * eta(q^18)^9 / (eta(q^2) * eta(q^3)^2 * eta(q^9)^3 * eta(q^12)^2 * eta(q^36)^3) in powers of q.
%F A164616 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 A164616 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 A258108. - _Michael Somos_, May 20 2015
%F A164616 a(3*n + 2) = 0. a(3*n) = A164617(n). a(3*n + 1) = -A132977(n).
%F A164616 Convolution inverse of A164615.
%F A164616 a(n) = (-1)^n * A258100(n). - _Michael Somos_, May 20 2015
%e A164616 G.f. = 1 - q + q^3 - 2*q^4 + 4*q^6 - 5*q^7 + 10*q^9 - 12*q^10 + 20*q^12 + ...
%t A164616 eta[x_] := QPochhammer[x]; A164616[n_] := SeriesCoefficient[eta[q]* eta[q^4]*eta[q^18]^9/(eta[q^2]*eta[q^3]^2*eta[q^9]^3*eta[q^12]^2* eta[q^36]^3), {q, 0, n}]; Table[A164616[n], {n,0,50}] (* _G. C. Greubel_, Aug 10 2017 *)
%o A164616 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^9 / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^9 + A)^3 * eta(x^12 + A)^2 * eta(x^36 + A)^3), n))};
%Y A164616 Cf. A132977, A164615, A164617, A258100, A258108.
%K A164616 sign
%O A164616 0,5
%A A164616 _Michael Somos_, Aug 17 2009