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A297331 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (theta_3(q^(1/2))^k + theta_4(q^(1/2))^k)/2.

%I A297331 #10 Apr 22 2018 05:10:39
%S A297331 1,1,0,1,0,0,1,4,2,0,1,12,4,0,0,1,24,6,0,0,0,1,40,24,24,4,0,0,1,60,90,
%T A297331 96,12,8,0,0,1,84,252,240,24,24,0,0,0,1,112,574,544,200,144,8,0,2,0,1,
%U A297331 144,1136,1288,1020,560,96,48,4,0,0,1,180,2034,3136,3444,1560,400,192,6,4,0,0
%N A297331 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (theta_3(q^(1/2))^k + theta_4(q^(1/2))^k)/2.
%H A297331 J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1007/978-1-4757-2016-7">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, p. 118.
%F A297331 G.f. of column k: (theta_3(q^(1/2))^k + theta_4(q^(1/2))^k)/2, where theta_() is the Jacobi theta function.
%e A297331 Square array begins:
%e A297331 1,  1,  1,   1,    1,    1,  ...
%e A297331 0,  0,  4,  12,   24,   40,  ...
%e A297331 0,  2,  4,   6,   24,   90,  ...
%e A297331 0,  0,  0,  24,   96,  240,  ...
%e A297331 0,  0,  4,  12,   24,  200,  ...
%e A297331 0,  0,  8,  24,  144,  560,  ...
%t A297331 Table[Function[k, SeriesCoefficient[(EllipticTheta[3, 0, q^(1/2)]^k + EllipticTheta[4, 0, q^(1/2)]^k)/2, {q, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%Y A297331 Columns k=0..32 give A000007, A089798 (absolute values), A004018, A004015, A004011, A005930, A008428, A008429, A008430, A008431, A008432, A022042, A022043, A022044, A022045, A022046, A022047, A022048, A022049, A022050, A022051, A022052, A022053, A022054, A022055, A022056, A022057, A022058, A022059, A022060, A022061, A022062, A022063.
%Y A297331 Main diagonal gives A303333.
%Y A297331 Cf. A122141, A286815.
%K A297331 nonn,tabl
%O A297331 0,8
%A A297331 _Ilya Gutkovskiy_, Dec 28 2017