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 A302997 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k/(1 - x), where theta_3() is the Jacobi theta function. 21
 1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 33, 29, 9, 1, 1, 11, 89, 123, 49, 11, 1, 1, 13, 221, 425, 257, 81, 13, 1, 1, 15, 485, 1343, 1281, 515, 113, 15, 1, 1, 17, 953, 4197, 5913, 3121, 925, 149, 17, 1, 1, 19, 1713, 12435, 23793, 16875, 6577, 1419, 197, 19, 1, 1, 21, 2869, 33809, 88273, 84769, 42205, 11833, 2109, 253, 21, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS A(n,k) is the number of integer lattice points inside the k-dimensional hypersphere of radius n. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1274 Eric Weisstein's World of Mathematics, Jacobi Theta Functions FORMULA A(n,k) = [x^(n^2)] (1/(1 - x))*(Sum_{j=-infinity..infinity} x^(j^2))^k. EXAMPLE Square array begins:   1,   1,   1,    1,     1,      1,  ...   1,   3,   5,    7,     9,     11,  ...   1,   5,  13,   33,    89,    221,  ...   1,   7,  29,  123,   425,   1343,  ...   1,   9,  49,  257,  1281,   5913,  ...   1,  11,  81,  515,  3121,  16875,  ... MATHEMATICA Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k/(1 - x), {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^2, {i, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten PROG (PARI) T(n, k)={if(k==0, 1, polcoef(((1 + 2*sum(j=1, n, x^(j^2)) + O(x*x^(n^2)))^k)/(1-x), n^2))} \\ Andrew Howroyd, Sep 14 2019 CROSSREFS Columns k=0..10 give A000012, A005408, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416. Rows k=0..10 give A000012, A005408, A055426, A055427, A055428, A055429, A055430, A055431, A055432, A055433, A055434. Main diagonal gives A302861. Cf. A000122, A122510, A302996, A302998. Sequence in context: A026714 A008288 A238339 * A326792 A144461 A106597 Adjacent sequences:  A302994 A302995 A302996 * A302998 A302999 A303000 KEYWORD nonn,tabl AUTHOR Ilya Gutkovskiy, Apr 17 2018 STATUS approved

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Last modified July 24 03:29 EDT 2021. Contains 346273 sequences. (Running on oeis4.)