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A257655 Expansion of f(x^3, x^9) * f(x^6, x^6) / f(-x, -x^2) in powers of x where f(,) is Ramanujan's general theta function.

%I A257655 #18 Feb 16 2025 08:33:25
%S A257655 1,1,2,4,6,9,16,22,33,50,70,98,138,188,256,348,463,614,812,1060,1378,
%T A257655 1785,2292,2932,3740,4736,5978,7522,9416,11750,14620,18116,22384,
%U A257655 27585,33878,41500,50714,61794,75120,91118,110247,133110,160390,192836,231400,277162
%N A257655 Expansion of f(x^3, x^9) * f(x^6, x^6) / f(-x, -x^2) in powers of x where f(,) is Ramanujan's general theta function.
%C A257655 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A257655 G. C. Greubel, <a href="/A257655/b257655.txt">Table of n, a(n) for n = 0..2500</a>
%H A257655 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A257655 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A257655 Expansion of f(-x, -x^5) * f(x^6, x^6) / f(-x, -x) in powers of x where f(,) is Ramanujan's general theta function.
%F A257655 Expansion of q^(-1/3) * eta(q^12)^5 / (eta(q) * eta(q^3) * eta(q^24)^2) in powers of q.
%F A257655 Euler transform of period 24 sequence [ 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, -3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, -1, ...].
%F A257655 a(n) = A097196(2*n).
%e A257655 G.f. = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + 9*x^5 + 16*x^6 + 22*x^7 + 33*x^8 + ...
%e A257655 G.f. = q + q^4 + 2*q^7 + 4*q^10 + 6*q^13 + 9*q^16 + 16*q^19 + 22*q^22 + ...
%t A257655 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^6] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8) QPochhammer[ x]), {x, 0, n}];
%t A257655 eta[q_] := q^(1/24)*QPochhammer[q]; With[{nmax = 50}, CoefficientList[ Series[q^(-1/3)*eta[q^12]^5/(eta[q]*eta[q^3]*eta[q^24]^2), {x, 0, nmax}], x]] (* _G. C. Greubel_, Aug 02 2018 *)
%o A257655 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^12 + A)^5 / (eta(x + A) * eta(x^3 + A) * eta(x^24 + A)^2), n))};
%Y A257655   Cf. A097196.
%K A257655 nonn
%O A257655 0,3
%A A257655 _Michael Somos_, Jul 25 2015