A187784 - OEIS

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

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OFFSET

0,6

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.

LINKS

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

FORMULA

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

EXAMPLE

: 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;

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

CROSSREFS

Cf. A000142, A000670, A032032.

Sequence in context: A091615 A213279 A175591 * A370901 A198812 A155183

Adjacent sequences: A187781 A187782 A187783 * A187785 A187786 A187787

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Jan 05 2013

STATUS

approved