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 A289522 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=0} ((1 + x^(2*j+1))/(1 - x^(2*j+1)))^k. 2
 1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 8, 4, 0, 1, 8, 18, 16, 6, 0, 1, 10, 32, 44, 32, 8, 0, 1, 12, 50, 96, 102, 56, 12, 0, 1, 14, 72, 180, 256, 216, 96, 16, 0, 1, 16, 98, 304, 550, 624, 428, 160, 22, 0, 1, 18, 128, 476, 1056, 1512, 1408, 816, 256, 30, 0, 1, 20, 162, 704, 1862, 3240, 3820, 3008, 1494, 404, 40, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Table of n, a(n) for n=0..77. FORMULA G.f. of column k: Product_{j>=0} ((1 + x^(2*j+1))/(1 - x^(2*j+1)))^k. G.f. of column 2k: (theta_3(x)/theta_4(x))^k, where theta_() is the Jacobi theta function. For asymptotics of column k see comment from Vaclav Kotesovec in A261648. EXAMPLE Square array begins: 1, 1, 1, 1, 1, 1, ... 0, 2, 4, 6, 8, 10, ... 0, 2, 8, 18, 32, 50, ... 0, 4, 16, 44, 96, 180, ... 0, 6, 32, 102, 256, 550, ... 0, 8, 56, 216, 624, 1512, ... MATHEMATICA Table[Function[k, SeriesCoefficient[Product[((1 + x^(2 i + 1))/(1 - x^(2 i + 1)))^k, {i, 0, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten Table[Function[k, SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten CROSSREFS Columns k=0-6 give: A000007, A080054, A007096, A261647, A014969, A261648, A014970. Rows n=0-3 give: A000012, A005843, A001105, A217873. Main diagonal gives A291697. Sequence in context: A194686 A302996 A266213 * A361397 A316273 A124915 Adjacent sequences: A289519 A289520 A289521 * A289523 A289524 A289525 KEYWORD nonn,tabl AUTHOR Ilya Gutkovskiy, Jul 07 2017 STATUS approved

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Last modified July 15 19:27 EDT 2024. Contains 374334 sequences. (Running on oeis4.)