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 A302996 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k, where theta_3() is the Jacobi theta function. 2
 1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 4, 2, 0, 1, 8, 6, 4, 2, 0, 1, 10, 24, 30, 4, 2, 0, 1, 12, 90, 104, 6, 12, 2, 0, 1, 14, 252, 250, 24, 30, 4, 2, 0, 1, 16, 574, 876, 730, 248, 30, 4, 2, 0, 1, 18, 1136, 3542, 4092, 1210, 312, 54, 4, 2, 0, 1, 20, 2034, 12112, 18494, 7812, 2250, 456, 6, 4, 2, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS A(n,k) is the number of ordered ways of writing n^2 as a sum of k squares. LINKS Eric Weisstein's World of Mathematics, Jacobi Theta Functions FORMULA A(n,k) = [x^(n^2)] (Sum_{j=-infinity..infinity} x^(j^2))^k. EXAMPLE Square array begins: 1,  1,   1,   1,    1,     1,  ... 0,  2,   4,   6,    8,    10,  ... 0,  2,   4,   6,   24,    90,  ... 0,  2,   4,  30,  104,   250,  ... 0,  2,   4,   6,   24,   730,  ... 0,  2,  12,  30,  248,  1210,  ... MATHEMATICA Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten Table[Function[k, SeriesCoefficient[Sum[x^i^2, {i, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten CROSSREFS Columns k=0..4 give A000007, A040000, A046109, A016725, A267326. Main diagonal gives A232173. Cf. A000122, A122141, A255212, A286815. Sequence in context: A228924 A246862 A194686 * A266213 A289522 A316273 Adjacent sequences:  A302993 A302994 A302995 * A302997 A302998 A302999 KEYWORD nonn,tabl AUTHOR Ilya Gutkovskiy, Apr 17 2018 STATUS approved

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Last modified October 14 07:31 EDT 2019. Contains 327995 sequences. (Running on oeis4.)