A108947 Triangle: T(n,k) is the partition function G(n-k,k).

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 4, 2, 1, 1, 0, 1, 10, 5, 2, 1, 1, 0, 1, 26, 14, 5, 2, 1, 1, 0, 1, 76, 46, 15, 5, 2, 1, 1, 0, 1, 232, 166, 51, 15, 5, 2, 1, 1, 0, 1, 764, 652, 196, 52, 15, 5, 2, 1, 1, 0, 1, 2620, 2780, 827, 202, 52, 15, 5, 2, 1, 1

Offset: 0

Author

Paul Boddington, Jul 21 2005

Keywords

• easy
• nonn
• tabl

Comments

See entries for A001680 and A001681 for appropriate references.

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Cf. A000110. First differences of a sequence G(k, 0), G(k, 1), ... give a row of A080510 (e.g., 0, 1, 10, 14, 15, 15, ... gives 1, 9, 4, 1).

Programs

Maple

G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(G(n-i*j, i-1)*n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
T:= (n, k)-> G(n-k, k):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Sep 15 2013

Mathematica

G[n_, i_] := G[n, i] = If[n == 0, 1, If[i<1, 0, Sum[G[n-i*j, i-1]*n!/i!^j/(n-i*j)! /j!, {j, 0, n/i}]]]; T[n_, k_] := G[n-k, k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

Formula

E.g.f. for sequence G(0, k), G(1, k), ... is exp(x + (1/2)*x^2 + ... + (1/k!)*x^k).

Extensions

One term corrected by Alois P. Heinz, Sep 15 2013