A225213 Triangular array read by rows. T(n,k) is the number of cycles in the digraph representation of all functions f:{1,2,...,n}->{1,2,...,n} that have length k; 1<=k<=n.

1, 4, 1, 27, 9, 2, 256, 96, 32, 6, 3125, 1250, 500, 150, 24, 46656, 19440, 8640, 3240, 864, 120, 823543, 352947, 168070, 72030, 24696, 5880, 720, 16777216, 7340032, 3670016, 1720320, 688128, 215040, 46080, 5040

Offset: 1

Author

Geoffrey Critzer, May 01 2013

Keywords

• nonn
• tabl

Comments

Row sums = A190314(n)

Sum_{k=1..n} T(n,k)*k = A063169(n)

T(n,n) = (n-1)!

Column 1 = n^n = A000312

Column 2 = A081131

Examples

			1,
4,      1,
27,     9,      2,
256,    96,     32,     6,
3125,   1250,   500,    150,   24,
46656,  19440,  8640,   3240,  864,   120,
823543, 352947, 168070, 72030, 24696, 5880, 720
		

Programs

Mathematica

Table[Table[(j-1)!Binomial[n,j]n^(n-j),{j,1,n}],{n,1,8}]//Grid

Formula

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

E.g.f. for column k: A(x)^k/k * B(x) where A(x) is e.g.f. for A000169 and B(x) is e.g.f. for A000312.