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A258096 Expansion of psi(x^4) * phi(-x^4)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta function.

%I A258096 #9 Feb 16 2025 08:33:25
%S A258096 1,2,4,8,7,10,12,8,18,18,16,24,21,20,28,32,20,32,36,24,42,42,28,48,57,
%T A258096 36,52,40,36,58,60,56,48,66,48,72,74,42,80,80,61,82,72,56,90,96,64,72,
%U A258096 98,70,100,104,64,106,108,72,114,96,84,144,111,84,104,128
%N A258096 Expansion of psi(x^4) * phi(-x^4)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta function.
%C A258096 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A258096 G. C. Greubel, <a href="/A258096/b258096.txt">Table of n, a(n) for n = 0..1000</a>
%H A258096 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258096 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258096 Expansion of q^(-1/2) * eta(q^2) * eta(q^4)^7 / (eta(q)^2 * eta(q^8)^2) in powers of q.
%F A258096 Euler transform of period 8 sequence [ 2, 1, 2, -6, 2, 1, 2, -4, ...].
%F A258096 a(n) = (-1)^n * A209940(n) = (-1)^floor(n/2) * A113419(n) = (-1)^(n + floor(n/2)) * A113417(n).
%e A258096 G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 7*x^4 + 10*x^5 + 12*x^6 + 8*x^7 + ...
%e A258096 G.f. = q + 2*q^3 + 4*q^5 + 8*q^7 + 7*q^9 + 10*q^11 + 12*q^13 + 8*q^15 + ...
%t A258096 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4]^7 / (QPochhammer[ x]^2 QPochhammer[ x^8]^2), {x, 0, n}];
%t A258096 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^2] EllipticTheta[ 4, 0, x^4]^4 / (EllipticTheta[ 4, 0, x] 2 x^(1/2)), {x, 0, n}];
%o A258096 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^7 / (eta(x + A)^2 * eta(x^8 + A)^2), n))};
%Y A258096 Cf. A113417, A113419, A209940.
%K A258096 nonn
%O A258096 0,2
%A A258096 _Michael Somos_, May 19 2015