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.

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

Offset: 0

Author

Ilya Gutkovskiy, Jul 07 2017

Keywords

• nonn
• tabl

Examples

			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,  ...
		

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.

Programs

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

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.