A091438 Triangle a(n,k) of partitions of n objects of 2 colors, k of which are black and each part with at least one black object.

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 6, 7, 7, 1, 4, 8, 12, 12, 11, 1, 4, 10, 16, 21, 19, 15, 1, 5, 12, 23, 31, 36, 30, 22, 1, 5, 15, 28, 45, 55, 58, 45, 30, 1, 6, 17, 37, 60, 84, 94, 92, 67, 42, 1, 6, 20, 44, 80, 115, 147, 153, 140, 97, 56, 1, 7, 23, 55, 101, 161, 211, 249, 244, 211, 139, 77

Offset: 1

Author

Wouter Meeussen and Christian G. Bower, Jan 08 2004

Keywords

• nonn
• tabl

Comments

Number of ways to factor p^(n-k)*q^k where p and q are distinct primes and each factor is a multiple of q.

Examples

			  1;
  1, 2;
  1, 2, 3;
  1, 3, 4, 5;
  1, 3, 6, 7, 7; ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Row sums: A000219.

Main diagonal: A000041.

a(2n,n) gives A108457.

Cf. A054225.

Programs

Maple

b:= proc(n, i, j, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
      `if`(i<1 or k<1, 0, `if`(j<1, b(n, i-1, i-1, k),
       b(n, i, j-1, k)+`if`(i>n or j>k, 0, b(n-i, i, j, k-j)))))
    end:
a:= (n, k)->  b(n$2, k$2):
seq(seq(a(n,k), k=1..n), n=1..15);  # Alois P. Heinz, Mar 14 2015

Mathematica

b[n_, i_, j_, k_] := b[n, i, j, k] = If[n == 0, If[k == 0, 1, 0], If[i < 1 || k < 1, 0, If[j < 1, b[n, i - 1, i - 1, k], b[n, i, j - 1, k] + If[i > n || j > k, 0, b[n - i, i, j, k - j]]]]]; a[n_, k_] :=  b[n, n, k, k]; Table[a[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 10 2016, after Alois P. Heinz *)

Formula

G.f.: A(x,y) = Product_{i>=1, j=1..i} (1/(1-x^i*y^j)).