A286653 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^(k*j))/(1 - x^j).

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 3, 4, 3, 0, 1, 1, 2, 3, 4, 5, 4, 0, 1, 1, 2, 3, 5, 6, 7, 5, 0, 1, 1, 2, 3, 5, 6, 9, 9, 6, 0, 1, 1, 2, 3, 5, 7, 10, 12, 13, 8, 0, 1, 1, 2, 3, 5, 7, 10, 13, 16, 16, 10, 0, 1, 1, 2, 3, 5, 7, 11, 14, 19, 22, 22, 12, 0

Offset: 0

Author

Ilya Gutkovskiy, May 11 2017

Keywords

• nonn
• tabl

Comments

A(n,k) is the number of partitions of n in which no parts are multiples of k.

A(n,k) is also the number of partitions of n into at most k-1 copies of each part.

Examples

			Square array begins:
  1,  1,  1,  1,  1,  1,  ...
  0,  1,  1,  1,  1,  1,  ...
  0,  1,  2,  2,  2,  2,  ...
  0,  2,  2,  3,  3,  3,  ...
  0,  2,  4,  4,  5,  5,  ...
  0,  3,  5,  6,  6,  7,  ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened
• Eric Weisstein's World of Mathematics, Partition Function b_k
• Index entries for sequences related to partitions

Crossrefs

Columns k=1-13 give: A000007, A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546, A341714.

Main diagonal gives A000041.

Mirror of A061198.

Cf. A061199, A210485.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(k*i*(i+1)/2[0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
A:= (n, k)-> b(n$2, k-1)[1]:
seq(seq(A(n, 1+d-n), n=0..d), d=0..16);  # Alois P. Heinz, Oct 17 2018

Mathematica

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

Formula

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