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A283000 Expansion of chi(-x)^2 * chi(x^3)^2 * chi(-x^4) / chi(x^6) in powers of x where chi() is a Ramanujan theta function.

%I A283000 #13 Feb 16 2025 08:33:42
%S A283000 1,-2,1,0,-1,0,0,2,1,0,0,0,-1,-4,-1,0,2,0,1,6,-2,0,-1,0,2,-8,1,0,-3,0,
%T A283000 -1,12,4,0,2,0,-5,-18,-2,0,5,0,2,24,-6,0,-3,0,8,-32,4,0,-9,0,-4,44,10,
%U A283000 0,4,0,-12,-58,-6,0,15,0,7,76,-17,0,-7,0,19,-100
%N A283000 Expansion of chi(-x)^2 * chi(x^3)^2 * chi(-x^4) / chi(x^6) in powers of x where chi() is a Ramanujan theta function.
%C A283000 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700)
%H A283000 G. C. Greubel, <a href="/A283000/b283000.txt">Table of n, a(n) for n = 0..1000</a>
%H A283000 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A283000 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A283000 Expansion of phi(x^3) * chi(-x)^2 / f(x^4, x^8) in powers of x where phi(), chi(), f() are Ramanujan theta functions.
%F A283000 Expansion of f(-x, -x^5)^2 / (f(x^4, x^8) * f(x^6, x^18)) in powers of x where f(, ) is Ramanujan's general theta functions.
%F A283000 Expansion of q^(1/4) * eta(q)^2 * eta(q^4) * eta(q^6)^5 * eta(q^24) / (eta(q^2)^2 * eta(q^3)^2 *eta(q^8) * eta(q^12)^4) in powers of q.
%F A283000 Euler transform of period 24 sequence [-2, 0, 0, -1, -2, -3, -2, 0, 0, 0, -2, 0, -2, 0, 0, 0, -2, -3, -2, -1, 0, 0, -2, 0, ...].
%F A283000 G.f.: Product_{k>0} (1 - x^k + x^(2*k))^2 * (1 - x^(4*k) + x^(8*k)) / (1 + x^(6*k))^3.
%F A283000 a(n) = A134178(2*n + 1). a(6*n + 3) = a(6*n + 5) = 0.
%e A283000 G.f. = 1 - 2*x + x^2 - x^4 + 2*x^7 + x^8 - x^12 - 4*x^13 - x^14 + 2*x^16 + ...
%e A283000 G.f. = q^-1 - 2*q^3 + q^7 - q^15 + 2*q^27 + q^31 - q^47 - 4*q^51 - q^55 + ...
%t A283000 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ -x^3, x^6]^2 QPochhammer[ x^4, x^8] QPochhammer[ x^6, -x^6], {x, 0, n}];
%o A283000 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^5 * eta(x^24 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^8 + A) * eta(x^12 + A)^4), n))};
%o A283000 (PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q)^2*eta(q^4)*eta(q^6)^5*eta(q^24)/(eta(q^2)^2*eta(q^3)^2 *eta(q^8)*eta(q^12)^4))} \\ _Altug Alkan_, Mar 21 2018
%Y A283000 Cf. A134178.
%K A283000 sign
%O A283000 0,2
%A A283000 _Michael Somos_, Feb 26 2017