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A186924 Expansion of (phi(-q^3) / phi(-q))^2 in powers of q where phi is a Ramanujan theta function.

%I A186924 #44 Feb 16 2025 08:33:14
%S A186924 1,4,12,28,60,120,228,416,732,1252,2088,3408,5460,8600,13344,20424,
%T A186924 30876,46152,68268,100016,145224,209120,298800,423840,597108,835804,
%U A186924 1162824,1608508,2212896,3028632,4124664,5590976,7544604,10137264,13565016
%N A186924 Expansion of (phi(-q^3) / phi(-q))^2 in powers of q where phi is a Ramanujan theta function.
%C A186924 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A186924 G. C. Greubel, <a href="/A186924/b186924.txt">Table of n, a(n) for n = 0..1000</a>
%H A186924 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A186924 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A186924 Euler transform of period 6 sequence [ 4, 2, 0, 2, 4, 0, ...].
%F A186924 Expansion of (eta(q^2) * eta(q^3)^2 / (eta(q)^2 * eta(q^6)))^2 in powers of q.
%F A186924 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/3)  g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A058487.
%F A186924 Convolution square of A098151. a(n) = 4 * A187100(n) unless n=0.
%F A186924 Convolution inverse of A217771. - _Michael Somos_, Sep 05 2015
%F A186924 a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 10 2015
%e A186924 G.f. = 1 + 4*q + 12*q^2 + 28*q^3 + 60*q^4 + 120*q^5 + 228*q^6 + 416*q^7 + 732*q^8 + ...
%t A186924 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^2 / EllipticTheta[ 4, 0, q]^2, {q, 0, n}]; (* _Michael Somos_, Sep 05 2015 *)
%t A186924 nmax = 50; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(3*k))^2 / ((1-x^k)^2 * (1-x^(6*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 10 2015 *)
%o A186924 (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 + A)^2 * eta(x^6 + A)))^2, n))};
%Y A186924 Cf. A058487, A098151, A187100, A217771.
%K A186924 nonn
%O A186924 0,2
%A A186924 _Michael Somos_, Mar 05 2011