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

1,3

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.

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Row sums: A000219.

Main diagonal: A000041.

a(2n,n) gives A108457.

Cf. A054225.

Sequence in context: A215520 A026820 A330661 * A011794 A221640 A073300

Adjacent sequences: A091435 A091436 A091437 * A091439 A091440 A091441

Keyword

nonn,tabl

Author

Wouter Meeussen and Christian G. Bower, Jan 08 2004

Status

approved