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A260168 Expansion of f(-x)^6 * f(-x^3)^2 / phi(-x^3)^8 in powers of q where phi(), f() are Ramanujan theta functions.

%I A260168 #13 Feb 16 2025 08:33:26
%S A260168 1,-6,9,24,-114,126,262,-1044,999,1852,-6672,5868,10103,-34134,28341,
%T A260168 46336,-149400,118872,186926,-581412,447507,682340,-2062332,1545336,
%U A260168 2297737,-6782508,4970241,7236280,-20938728,15056694,21531158,-61246128,43329078,61003980
%N A260168 Expansion of f(-x)^6 * f(-x^3)^2 / phi(-x^3)^8 in powers of q where phi(), f() are Ramanujan theta functions.
%C A260168 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260168 G. C. Greubel, <a href="/A260168/b260168.txt">Table of n, a(n) for n = 0..1000</a>
%H A260168 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260168 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260168 Expansion of q^(-1/2) * (eta(q)^3 * eta(q^6)^4 / eta(q^3)^7)^2 in powers of q.
%F A260168 Euler transform of period 6 sequence [ -6, -6, 8, -6, -6, 0, ...].
%F A260168 -2 * a(n) = A261576(2*n + 1).
%e A260168 G.f. = 1 - 6*x + 9*x^2 + 24*x^3 - 114*x^4 + 126*x^5 + 262*x^6 - 1044*x^7 + ...
%e A260168 G.f. = q - 6*q^3 + 9*q^5 + 24*q^7 - 114*q^9 + 126*q^11 + 262*q^13 - 1044*q^15 + ...
%t A260168 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^6 QPochhammer[ x^3]^2 / EllipticTheta[ 4, 0, x^3]^8, {x, 0, n}];
%o A260168 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 * eta(x^6 + A)^4 / eta(x^3 + A)^7)^2, n))};
%Y A260168 Cf. A261576.
%K A260168 sign
%O A260168 0,2
%A A260168 _Michael Somos_, Nov 09 2015