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A288515
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Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.
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4
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1, 1, 0, 1, 2, 0, 1, 4, 4, 0, 1, 6, 12, 8, 0, 1, 8, 24, 32, 14, 0, 1, 10, 40, 80, 76, 24, 0, 1, 12, 60, 160, 234, 168, 40, 0, 1, 14, 84, 280, 552, 624, 352, 64, 0, 1, 16, 112, 448, 1110, 1712, 1552, 704, 100, 0, 1, 18, 144, 672, 2004, 3912, 4896, 3648, 1356, 154, 0, 1, 20, 180, 960, 3346, 7896, 12600, 13120, 8184, 2532, 232, 0
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OFFSET
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0,5
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LINKS
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Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to partitions
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FORMULA
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G.f. of column k: Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.
G.f. of column k: 1/theta_4(x)^k, where theta_4() is the Jacobi theta function.
For asymptotics of column k see comment from Vaclav Kotesovec in A001934.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 2, 4, 6, 8, 10, ...
0, 4, 12, 24, 40, 60, ...
0, 8, 32, 80, 160, 280, ...
0, 14, 76, 234, 552, 1110, ...
0, 24, 168, 624, 1712, 3913, ...
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MATHEMATICA
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Table[Function[k, SeriesCoefficient[Product[((1 + x^i)/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/EllipticTheta[4, 0, x]^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
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PROG
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(Julia) # JacobiTheta4 is defined in A002448.
A288515Column(k, len) = JacobiTheta4(len, -k)
for k in 0:8 A288515Column(k, 8) |> println end # Peter Luschny, Mar 12 2018
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CROSSREFS
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Columns k=0-24 give: A000007, A015128, A001934, A004404 (alternating values), A284286, A004406-A004425 (alternating values).
Rows n=0-2 give: A000012, A005843, A046092.
Main diagonal gives A270919.
Antidiagonal sums give A299108.
Cf. A122141, A286815.
Sequence in context: A209063 A342321 A098689 * A264583 A158984 A158417
Adjacent sequences: A288512 A288513 A288514 * A288516 A288517 A288518
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KEYWORD
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nonn,tabl
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AUTHOR
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Ilya Gutkovskiy, Jun 10 2017
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STATUS
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approved
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