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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.

%I A187784 #20 Sep 21 2016 10:49:34
%S A187784 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 A187784 1800,0,720,743,3584,5166,13440,4200,15120,0,5040,5699,22864,67368,
%U A187784 68544,159600,40320,141120,0,40320,42241,225684,502200,1161216,922320,1995840,423360,1451520,0,362880
%N 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.
%C A187784 A singleton is a set that contains exactly one element.
%C A187784 Column for k=0 is A032032.
%C A187784 Row sums are A000670.
%C A187784 Main diagonal is A000142.
%H A187784 Alois P. Heinz, <a href="/A187784/b187784.txt">Rows n = 0..140, flattened</a>
%F A187784 E.g.f.: 1/(2 - exp(x) + x - y*x).
%e A187784 :    1;
%e A187784 :    0,     1;
%e A187784 :    1,     0,     2;
%e A187784 :    1,     6,     0,     6;
%e A187784 :    7,     8,    36,     0,     24;
%e A187784 :   21,   100,    60,   240,      0,   120;
%e A187784 :  141,   372,  1170,   480,   1800,     0,    720;
%e A187784 :  743,  3584,  5166, 13440,   4200, 15120,      0, 5040;
%e A187784 : 5699, 22864, 67368, 68544, 159600, 40320, 141120,    0, 40320;
%p A187784 with(combinat):
%p A187784 b:= proc(n, i, p) option remember; `if`(n=0, p!,
%p A187784       `if`(i<2, 0, add(multinomial(n, n-i*j, i$j)
%p A187784       *b(n-i*j, i-1, p+j)/j!, j=0..n/i)))
%p A187784     end:
%p A187784 T:= (n, k)-> binomial(n, k)*b(n-k$2, k):
%p A187784 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Sep 06 2015
%t A187784 nn=8;Range[0,nn]!CoefficientList[Series[1/(2-Exp[x]+x-y x),{x,0,nn}],{x,y}]//Grid
%Y A187784 Cf. A000142, A000670, A032032.
%K A187784 nonn,tabl
%O A187784 0,6
%A A187784 _Geoffrey Critzer_, Jan 05 2013