A288515 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.

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

Offset: 0

Author

Ilya Gutkovskiy, Jun 10 2017

Keywords

• nonn
• tabl

Examples

			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,  ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened
• Index entries for sequences related to partitions

Crossrefs

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.

Programs

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

Mathematica

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

Formula

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.