A225723 Triangular array read by rows: T(n,k) is the number of size k components in the digraph representation of all functions f:{1,2,...,n}->{1,2,...,n}; n>=1, 1<=k<=n.

1, 2, 3, 12, 9, 17, 108, 72, 68, 142, 1280, 810, 680, 710, 1569, 18750, 11520, 9180, 8520, 9414, 21576, 326592, 196875, 152320, 134190, 131796, 151032, 355081, 6588344, 3919104, 2975000, 2544640, 2372328, 2416512, 2840648, 6805296

Offset: 1

Author

Geoffrey Critzer, May 13 2013

Keywords

• nonn
• tabl

Comments

T(n,1) = n*(n-1)^(n-1) = A055897(n).

Row sums = A190314.

T(n,n) = A001865(n).

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

Examples

			Triangle T(n,k) begins:
       1;
       2,      3;
      12,      9,     17;
     108,     72,     68,    142;
    1280,    810,    680,    710,   1569;
   18750,  11520,   9180,   8520,   9414,  21576;
  326592, 196875, 152320, 134190, 131796, 151032, 355081;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..100, flattened

Crossrefs

Cf. A225213.

Programs

Maple

b:= n-> n!*add(n^(n-k-1)/(n-k)!, k=1..n):
T:= (n, k)-> binomial(n,k)*b(k)*(n-k)^(n-k):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, May 13 2013

Mathematica

nn = 8; tx = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; txy =
Sum[n^(n - 1) (x y)^n/n!, {n, 1, nn}];
Map[Select[#, # > 0 &] &,
  Drop[Range[0, nn]! CoefficientList[
     Series[Log[1/(1 - txy)]/(1 - tx), {x, 0, nn}], {x, y}],
   1]] // Grid

Formula

E.g.f.: log(1/(1 - A(x*y)))/(1 - A(x)) where A(x) is the e.g.f. for A000169.

T(n,k) = C(n,k)*A001865(k)*A000312(n-k). - Alois P. Heinz, May 13 2013