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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.
3

%I #20 Sep 21 2016 10:49:34

%S 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,

%T 1800,0,720,743,3584,5166,13440,4200,15120,0,5040,5699,22864,67368,

%U 68544,159600,40320,141120,0,40320,42241,225684,502200,1161216,922320,1995840,423360,1451520,0,362880

%N 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.

%C A singleton is a set that contains exactly one element.

%C Column for k=0 is A032032.

%C Row sums are A000670.

%C Main diagonal is A000142.

%H Alois P. Heinz, <a href="/A187784/b187784.txt">Rows n = 0..140, flattened</a>

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

%e : 1;

%e : 0, 1;

%e : 1, 0, 2;

%e : 1, 6, 0, 6;

%e : 7, 8, 36, 0, 24;

%e : 21, 100, 60, 240, 0, 120;

%e : 141, 372, 1170, 480, 1800, 0, 720;

%e : 743, 3584, 5166, 13440, 4200, 15120, 0, 5040;

%e : 5699, 22864, 67368, 68544, 159600, 40320, 141120, 0, 40320;

%p with(combinat):

%p b:= proc(n, i, p) option remember; `if`(n=0, p!,

%p `if`(i<2, 0, add(multinomial(n, n-i*j, i$j)

%p *b(n-i*j, i-1, p+j)/j!, j=0..n/i)))

%p end:

%p T:= (n, k)-> binomial(n, k)*b(n-k$2, k):

%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Sep 06 2015

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

%Y Cf. A000142, A000670, A032032.

%K nonn,tabl

%O 0,6

%A _Geoffrey Critzer_, Jan 05 2013