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A164614 Expansion of (chi(q) / chi^3(q^3))^2 in powers of q where chi() is a Ramanujan theta function.

%I A164614 #12 Feb 16 2025 08:33:11
%S A164614 1,2,1,-4,-8,-2,14,24,6,-38,-63,-16,92,150,36,-208,-329,-78,440,684,
%T A164614 160,-884,-1358,-312,1710,2592,590,-3196,-4796,-1082,5800,8632,1929,
%U A164614 -10270,-15162,-3364,17784,26078,5750,-30192,-44010,-9644,50369,73012,15916,-82698,-119280,-25880,133818
%N A164614 Expansion of (chi(q) / chi^3(q^3))^2 in powers of q where chi() is a Ramanujan theta function.
%C A164614 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A164614 G. C. Greubel, <a href="/A164614/b164614.txt">Table of n, a(n) for n = 0..1000</a>
%H A164614 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A164614 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A164614 Expansion of q^(-2/3) * (eta(q^2)^2 * eta(q^3)^3 * eta(q^12)^3 / (eta(q) * eta(q^4) * eta(q^6)^6))^2 in powers of q.
%F A164614 Euler transform of period 12 sequence [ 2, -2, -4, 0, 2, 4, 2, 0, -4, -2, 2, 0, ...].
%F A164614 Convolution square of A128111.
%e A164614 G.f. = 1 + 2*x + x^2 - 4*x^3 - 8*x^4 - 2*x^5 + 14*x^6 + 24*x^7 + 6*x^8 + ...
%e A164614 G.f. = q^2 + 2*q^5 + q^8 - 4*q^11 - 8*q^14 - 2*q^17 + 14*q^20 + 24*q^23 + ...
%t A164614 a[ n_] := SeriesCoefficient[(QPochhammer[-q, q^2]/(QPochhammer[-q^3, q^6])^3)^2, {q, 0, n}]; (* _Michael Somos_, Sep 02 2015 *)(* Corrected by G. C. Greubel_, Aug 10 2017 *)
%o A164614 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^12 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^6))^2, n))};
%Y A164614 Cf. A128111.
%K A164614 sign
%O A164614 0,2
%A A164614 _Michael Somos_, Aug 17 2009