login
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

%I #6 May 02 2013 06:24:17

%S 1,4,1,27,9,2,256,96,32,6,3125,1250,500,150,24,46656,19440,8640,3240,

%T 864,120,823543,352947,168070,72030,24696,5880,720,16777216,7340032,

%U 3670016,1720320,688128,215040,46080,5040

%N 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.

%C Row sums = A190314(n)

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

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

%C Column 1 = n^n = A000312

%C Column 2 = A081131

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

%F 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.

%e 1,

%e 4, 1,

%e 27, 9, 2,

%e 256, 96, 32, 6,

%e 3125, 1250, 500, 150, 24,

%e 46656, 19440, 8640, 3240, 864, 120,

%e 823543, 352947, 168070, 72030, 24696, 5880, 720

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

%K nonn,tabl

%O 1,2

%A _Geoffrey Critzer_, May 01 2013