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.

1, 1, 2, 3, 4, 9, 16, 18, 27, 64, 125, 128, 162, 256, 625, 1296, 1250, 1440, 1920, 3125, 7776, 16807, 15552, 16875, 20480, 28125, 46656, 117649, 262144, 235298, 244944, 280000, 350000, 489888, 823543, 2097152

Offset: 1

Author

Geoffrey Critzer, Nov 17 2013

Keywords

• nonn
• tabl

Comments

Column 1 is A000272.

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

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

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

Examples

			1;
1,    2;
3,    4,    9;
16,   18,   27,   64;
125,  128,  162,  256,  625;
1296, 1250, 1440, 1920, 3125, 7776;
		

References

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

Programs

Mathematica

Table[Table[Binomial[n,k](k+1)(k+1)^(k-1)(n-k+1)^(n-k-1),{k,0,n}],{n,0,7}]//Grid

Formula

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