A286957 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + k*x^j).

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 2, 0, 1, 4, 3, 6, 2, 0, 1, 5, 4, 12, 6, 3, 0, 1, 6, 5, 20, 12, 10, 4, 0, 1, 7, 6, 30, 20, 21, 18, 5, 0, 1, 8, 7, 42, 30, 36, 48, 22, 6, 0, 1, 9, 8, 56, 42, 55, 100, 57, 30, 8, 0, 1, 10, 9, 72, 56, 78, 180, 116, 84, 42, 10, 0, 1, 11, 10, 90, 72, 105, 294, 205, 180, 120, 66, 12, 0

Offset: 0

Author

Ilya Gutkovskiy, May 17 2017

Keywords

• nonn
• tabl

Comments

A(n,k) is the number of partitions of n into distinct parts of k sorts: the parts are unordered, but not the sorts.

Examples

			Square array begins:
1,  1,   1,   1,   1,   1,  ...
0,  1,   2,   3,   4,   5,  ...
0,  1,   2,   3,   4,   5,  ...
0,  2,   6,  12,  20,  30,  ...
0,  2,   6,  12,  20,  30,  ...
0,  3,  10,  21,  36,  55,  ...
		

Links

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

Crossrefs

Columns k=0-5 give: A000007, A000009, A032302, A032308, A261568, A261569.

Main diagonal gives A291698.

Cf. A246935.

Programs

Mathematica

Table[Function[k, SeriesCoefficient[Product[(1 + k x^i), {i, 1, Infinity}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-k, x]/(1 + k), {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

Formula

G.f. of column k: Product_{j>=1} (1 + k*x^j).