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A232055 Triangular array read by rows: T(n,k) is the number of forests of rooted labeled trees such that the vertex labeled with 1 is in a component (rooted tree) of size k, n>=1, 1<=k<=n.

%I A232055 #14 Nov 19 2013 08:44:22
%S A232055 1,1,2,3,4,9,16,18,27,64,125,128,162,256,625,1296,1250,1440,1920,3125,
%T A232055 7776,16807,15552,16875,20480,28125,46656,117649,262144,235298,244944,
%U A232055 280000,350000,489888,823543,2097152
%N A232055 Triangular array read by rows: T(n,k) is the number of forests of rooted labeled trees such that the vertex labeled with 1 is in a component (rooted tree) of size k, n>=1, 1<=k<=n.
%C A232055 Column 1 is A000272.
%C A232055 T(n,n) = A000169(n).
%C A232055 T(n+1,n) = A000312(n).
%C A232055 T(n+2,n)/3 = A081132(n-1).
%D A232055 Miklos Bona, Introduction to Enumerative Combinatorics, McGraw Hill, 2007, page 282.
%F A232055 T(n,k) = binomial(n-1,k-1)*k^(k-1)*(n-k+1)^(n-k-1).
%e A232055 1;
%e A232055 1,    2;
%e A232055 3,    4,    9;
%e A232055 16,   18,   27,   64;
%e A232055 125,  128,  162,  256,  625;
%e A232055 1296, 1250, 1440, 1920, 3125, 7776;
%t A232055 Table[Table[Binomial[n,k](k+1)(k+1)^(k-1)(n-k+1)^(n-k-1),{k,0,n}],{n,0,7}]//Grid
%K A232055 nonn,tabl
%O A232055 1,3
%A A232055 _Geoffrey Critzer_, Nov 17 2013