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%I #5 Feb 22 2019 05:16:56
%S 1,1,2,1,2,0,1,2,2,0,1,2,0,4,2,1,2,2,2,2,0,1,2,0,4,6,0,0,1,2,2,0,4,0,
%T 4,0,1,2,0,6,2,4,0,0,0,1,2,2,0,6,2,8,4,2,2,1,2,0,4,2,4,4,8,0,6,0,1,2,
%U 2,2,4,0,14,0,6,2,0,0,1,2,0,4,6,4,0,8,0,6,0,4,0,1,2,2,0,2,0,8,2,6,6,8,0,4,0
%N Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{d|k} theta_3(q^d).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F G.f. of column k: Product_{d|k} theta_3(q^d).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 2, 2, 2, 2, 2, 2, ...
%e 0, 2, 0, 2, 0, 2, ...
%e 0, 4, 2, 4, 0, 6, ...
%e 2, 2, 6, 4, 2, 6, ...
%e 0, 0, 0, 4, 2, 4, ...
%t Table[Function[k, SeriesCoefficient[Product[EllipticTheta[3, 0, q^d], {d, Divisors[k]}], {q, 0, n}]][i - n + 1], {i, 0, 13}, {n, 0, i}] // Flatten
%Y Columns k=1..48 give A000122, A033715, A033716, A033717, A033718, A033712, A033719, A033720, A033721, A033722, A033723, A033724, A033725, A033726, A033727, A033728, A033729, A033730, A033731, A033732, A033733, A033734, A033735, A033736, A033737, A033738, A033739, A033740, A033741, A033742, A033743, A033744, A033745, A033746, A033747, A033748, A033749, A033750, A033751, A033752, A033753, A033754, A033755, A033756, A033757, A033758, A033759, A033760.
%Y Cf. A320305 (diagonal).
%K nonn,tabl
%O 0,3
%A _Ilya Gutkovskiy_, Feb 21 2019
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