A286335 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)^k.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 2, 0, 1, 5, 10, 13, 9, 3, 0, 1, 6, 15, 24, 24, 14, 4, 0, 1, 7, 21, 40, 51, 42, 22, 5, 0, 1, 8, 28, 62, 95, 100, 73, 32, 6, 0, 1, 9, 36, 91, 162, 206, 190, 120, 46, 8, 0, 1, 10, 45, 128, 259, 384, 425, 344, 192, 66, 10, 0

Offset: 0

Author

Ilya Gutkovskiy, May 07 2017

Keywords

• nonn
• tabl

Comments

A(n,k) is the number of partitions of n into distinct parts (or odd parts) with k types of each part.

Examples

			A(3,2) = 6 because we have [3], [3'], [2, 1], [2', 1], [2, 1'] and [2', 1'] (partitions of 3 into distinct parts with 2 types of each part).
Also A(3,2) = 6 because we have [3], [3'], [1, 1, 1], [1, 1, 1'], [1, 1', 1'] and [1', 1', 1'] (partitions of 3 into odd parts with 2 types of each part).
Square array begins:
  1,  1,  1,   1,   1,   1,  ...
  0,  1,  2,   3,   4,   5,  ...
  0,  1,  3,   6,  10,  15,  ...
  0,  2,  6,  13,  24,  40,  ...
  0,  2,  9,  24,  51,  95,  ...
  0,  3, 14,  42, 100, 206,  ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened
• N. J. A. Sloane, Transforms

Crossrefs

Columns k=0-32 give: A000007, A000009, A022567-A022596.

Rows n=0-2 give: A000012, A001477, A000217.

Main diagonal gives A270913.

Antidiagonal sums give A299106.

Cf. A144064, A286352, A308680.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..n/i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 29 2019

Mathematica

Table[Function[k, SeriesCoefficient[Product[(1 + x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

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

A(n,k) = Sum_{i=0..k} binomial(k,i) * A308680(n,k-i). - Alois P. Heinz, Aug 29 2019