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A302998 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] (1 + theta_3(x))^k/(2^k*(1 - x)), where theta_3() is the Jacobi theta function.

%I A302998 #11 Feb 16 2025 08:33:53
%S A302998 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,5,11,11,5,1,1,6,20,29,17,6,1,1,7,36,
%T A302998 70,54,26,7,1,1,8,63,157,165,99,35,8,1,1,9,106,337,482,357,163,45,9,1,
%U A302998 1,10,171,702,1319,1203,688,239,58,10,1,1,11,265,1420,3390,3819,2673,1154,344,73,11,1
%N A302998 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] (1 + theta_3(x))^k/(2^k*(1 - x)), where theta_3() is the Jacobi theta function.
%C A302998 A(n,k) is the number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_k)^2 <= n^2.
%H A302998 Andrew Howroyd, <a href="/A302998/b302998.txt">Table of n, a(n) for n = 0..1274</a>
%H A302998 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A302998 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A302998 A(n,k) = [x^(n^2)] (1/(1 - x))*(Sum_{j>=0} x^(j^2))^k.
%e A302998 Square array begins:
%e A302998   1,  1,   1,   1,    1,     1,  ...
%e A302998   1,  2,   3,   4,    5,     6,  ...
%e A302998   1,  3,   6,  11,   20,    36,  ...
%e A302998   1,  4,  11,  29,   70,   157,  ...
%e A302998   1,  5,  17,  54,  165,   482,  ...
%e A302998   1,  6,  26,  99,  357,  1203,  ...
%t A302998 Table[Function[k, SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^k/(2^k (1 - x)), {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%t A302998 Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^2, {i, 0, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%o A302998 (PARI) T(n,k)={if(k==0, 1, polcoef(((sum(j=0, n, x^(j^2)) + O(x*x^(n^2)))^k)/(1-x), n^2))} \\ _Andrew Howroyd_, Sep 14 2019
%Y A302998 Columns k=0..10 give A000012, A000027, A000603, A000604, A055403, A055404, A055405, A055406, A055407, A055408, A055409.
%Y A302998 Rows n=0..10 give A000012, A000027, A055417, A055418, A055419, A055420, A055421, A055422, A055423, A055424, A055425.
%Y A302998 Main diagonal gives A302863.
%Y A302998 Cf. A000122, A122510, A302996, A302997.
%K A302998 nonn,tabl
%O A302998 0,5
%A A302998 _Ilya Gutkovskiy_, Apr 17 2018