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A219034 Triangular array read by rows: T(n,k) is the number of forests of rooted trees on n labeled nodes with exactly k isolated nodes; n>=0, 0<=k<=n.

%I A219034 #16 Dec 31 2021 14:58:22
%S A219034 1,0,1,2,0,1,9,6,0,1,76,36,12,0,1,805,380,90,20,0,1,10626,4830,1140,
%T A219034 180,30,0,1,167839,74382,16905,2660,315,42,0,1,3091768,1342712,297528,
%U A219034 45080,5320,504,56,0,1,65127465,27825912,6042204,892584,101430,9576,756,72,0,1
%N A219034 Triangular array read by rows: T(n,k) is the number of forests of rooted trees on n labeled nodes with exactly k isolated nodes; n>=0, 0<=k<=n.
%C A219034 Column k=0 is A105785.
%C A219034 Row sums = (n+1)^(n-1).
%H A219034 Alois P. Heinz, <a href="/A219034/b219034.txt">Rows n = 0..140, flattened</a>
%F A219034 E.g.f.: exp(T(x)-x+y*x) where T(x) is the e.g.f. for A000169.
%e A219034 Triangle T(n,k) begins:
%e A219034         1;
%e A219034         0,       1;
%e A219034         2,       0,      1;
%e A219034         9,       6,      0,     1;
%e A219034        76,      36,     12,     0,    1;
%e A219034       805,     380,     90,    20,    0,   1;
%e A219034     10626,    4830,   1140,   180,   30,   0,  1;
%e A219034    167839,   74382,  16905,  2660,  315,  42,  0, 1;
%e A219034   3091768, 1342712, 297528, 45080, 5320, 504, 56, 0, 1;
%e A219034   ...
%p A219034 b:= proc(n) option remember; expand(`if`(n=0, 1, add(i^(i-1)
%p A219034       *b(n-i)*binomial(n-1, i-1)*`if`(i=1, x, 1), i=1..n)))
%p A219034     end:
%p A219034 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
%p A219034 seq(T(n), n=0..10);  # _Alois P. Heinz_, Dec 31 2021
%t A219034 nn=8; t=Sum[n^(n-1)x^n/n!, {n,1,nn}]; Range[0,nn]! CoefficientList[ Series[Exp[t-x+y x], {x,0,nn}], {x,y}] //Grid
%K A219034 nonn,tabl
%O A219034 0,4
%A A219034 _Geoffrey Critzer_, Nov 10 2012