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A089813 Expansion of Jacobi theta function (theta_2(q) - 3*theta_2(q^9))/(2 q^(1/4)) in powers of q.

%I A089813 #20 Feb 16 2025 08:32:51
%S A089813 1,0,-2,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,1,0,0,
%T A089813 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,
%U A089813 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0
%N A089813 Expansion of Jacobi theta function (theta_2(q) - 3*theta_2(q^9))/(2 q^(1/4)) in powers of q.
%C A089813 This is the sequence A089812 with interleaved zeros. - _Michael Somos_, Nov 21 2017
%H A089813 G. C. Greubel, <a href="/A089813/b089813.txt">Table of n, a(n) for n = 0..5000</a>
%H A089813 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A089813 I. J. Zucker, <a href="https://doi.org/10.1088/0305-4470/23/2/009">Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums</a>, J. Phys. A: Math. Gen. 23, 117-132, 1990.
%F A089813 Euler transform of period 12 sequence [0, -2, 0, -1, 0, -1, 0, -1, 0, -2, 0, -1, ...]. - _Michael Somos_, Nov 21 2017
%e A089813 G.f. = 1 - 2*x^2 + x^6 + x^12 - 2*x^20 + x^30 + x^42 - 2*x^56 + x^72 + x^90 - 2*x^110 + ...
%e A089813 G.f. = q - 2*q^9 + q^25 + q^49 - 2*q^81 + q^121 + q^169 - 2*q^225 + q^289 + q^361 + ...
%t A089813 A089813[n_] := SeriesCoefficient[(EllipticTheta[2, 0, q] - 3*EllipticTheta[2, 0, q^9])/(2 q^(1/4), {q, 0, n}]; Table[A089813[n], {n, 0, 50}] (* _G. C. Greubel_, Nov 20 2017 *)
%t A089813 a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^2 QPochhammer[ q^12] / (QPochhammer[q ^4] QPochhammer[ q^6]), {q, 0, n}]; (* _Michael Somos_, Nov 21 2017 *)
%o A089813 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A) / (eta(x^4 + A) * eta(x^6 + A)), n))}; /* _Michael Somos_, Nov 21 2017 */
%o A089813 (PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^2)^2*eta(q^12)/(eta(q^4)*eta(q^6)))} \\ _Altug Alkan_, Mar 22 2018
%Y A089813 Cf. A089812.
%K A089813 sign
%O A089813 0,3
%A A089813 _Eric W. Weisstein_, Nov 12 2003