A187784 Triangular array read by rows: T(n,k) is the number of ordered set partitions of {1,2,...,n} with exactly k singletons, n>=0, 0<=k<=n.

1, 0, 1, 1, 0, 2, 1, 6, 0, 6, 7, 8, 36, 0, 24, 21, 100, 60, 240, 0, 120, 141, 372, 1170, 480, 1800, 0, 720, 743, 3584, 5166, 13440, 4200, 15120, 0, 5040, 5699, 22864, 67368, 68544, 159600, 40320, 141120, 0, 40320, 42241, 225684, 502200, 1161216, 922320, 1995840, 423360, 1451520, 0, 362880

Offset: 0

Author

Geoffrey Critzer, Jan 05 2013

Keywords

• nonn
• tabl

Comments

A singleton is a set that contains exactly one element.

Column for k=0 is A032032.

Row sums are A000670.

Main diagonal is A000142.

Examples

			:    1;
:    0,     1;
:    1,     0,     2;
:    1,     6,     0,     6;
:    7,     8,    36,     0,     24;
:   21,   100,    60,   240,      0,   120;
:  141,   372,  1170,   480,   1800,     0,    720;
:  743,  3584,  5166, 13440,   4200, 15120,      0, 5040;
: 5699, 22864, 67368, 68544, 159600, 40320, 141120,    0, 40320;
		

Links

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

Crossrefs

Cf. A000142, A000670, A032032.

Programs

Maple

with(combinat):
b:= proc(n, i, p) option remember; `if`(n=0, p!,
      `if`(i<2, 0, add(multinomial(n, n-i*j, i$j)
      *b(n-i*j, i-1, p+j)/j!, j=0..n/i)))
    end:
T:= (n, k)-> binomial(n, k)*b(n-k$2, k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Sep 06 2015

Mathematica

nn=8;Range[0,nn]!CoefficientList[Series[1/(2-Exp[x]+x-y x),{x,0,nn}],{x,y}]//Grid

Formula

E.g.f.: 1/(2 - exp(x) + x - y*x).