A232055 - OEIS

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 #14 Nov 19 2013 08:44:22

%S 1,1,2,3,4,9,16,18,27,64,125,128,162,256,625,1296,1250,1440,1920,3125,

%T 7776,16807,15552,16875,20480,28125,46656,117649,262144,235298,244944,

%U 280000,350000,489888,823543,2097152

%N 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 Column 1 is A000272.

%C T(n,n) = A000169(n).

%C T(n+1,n) = A000312(n).

%C T(n+2,n)/3 = A081132(n-1).

%D Miklos Bona, Introduction to Enumerative Combinatorics, McGraw Hill, 2007, page 282.

%F T(n,k) = binomial(n-1,k-1)*k^(k-1)*(n-k+1)^(n-k-1).

%e 1;

%e 1, 2;

%e 3, 4, 9;

%e 16, 18, 27, 64;

%e 125, 128, 162, 256, 625;

%e 1296, 1250, 1440, 1920, 3125, 7776;

%t 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 nonn,tabl

%O 1,3

%A _Geoffrey Critzer_, Nov 17 2013