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A123884 Expansion of phi(x) * phi(-x^6) / chi(-x^2) in powers of x where phi(), chi() are Ramanujan theta functions.

%I A123884 #12 Feb 16 2025 08:33:03
%S A123884 1,2,1,2,3,2,2,0,2,2,1,4,0,2,3,2,2,0,4,2,2,0,0,2,1,4,2,2,2,2,3,2,0,2,
%T A123884 2,2,2,0,2,4,4,0,0,0,1,2,4,0,2,4,2,2,1,6,0,2,2,0,0,2,4,2,0,2,2,0,4,0,
%U A123884 4,2,1,2,0,2,4,0,0,2,2,4,3,4,0,2,2,2,2
%N A123884 Expansion of phi(x) * phi(-x^6) / chi(-x^2) in powers of x where phi(), chi() are Ramanujan theta functions.
%C A123884 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A123884 G. C. Greubel, <a href="/A123884/b123884.txt">Table of n, a(n) for n = 0..1000</a>
%H A123884 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A123884 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A123884 Expansion of q^(-1/12) * eta(q^2)^4 * eta(q^6)^2 / (eta(q)^2 * eta(q^4) * eta(q^12)) in powers of q.
%F A123884 Euler transform of period 12 sequence [ 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -2, ...].
%F A123884 a(n) = A093829(12*n + 1).
%F A123884 a(n) = (-1)^n * A248886(n). a(2*n) = A131961(n). a(2*n + 1) = 2 * A131963(n). - _Michael Somos_, Oct 01 2015
%e A123884 G.f. = 1 + 2*x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + ...
%e A123884 G.f. = q + 2*q^13 + q^25 + 2*q^37 + 3*q^49 + 2*q^61 + 2*q^73 + 2*q^97 + 2*q^109 + ...
%t A123884 a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^2 EllipticTheta[ 4, 0, x^6] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* _Michael Somos_, Oct 01 2015 *)
%t A123884 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x^2, x^2], {x, 0, n}]; (* _Michael Somos_, Oct 01 2015 *)
%o A123884 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^2 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^12 + A)), n))};
%Y A123884 Cf. A093829, A131961, A131962, A248886.
%K A123884 nonn
%O A123884 0,2
%A A123884 _Michael Somos_, Oct 17 2006