A349454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, all of which are isolated; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 1, 0, 1, 8, 3, 0, 1, 81, 32, 6, 0, 1, 1024, 405, 80, 10, 0, 1, 15625, 6144, 1215, 160, 15, 0, 1, 279936, 109375, 21504, 2835, 280, 21, 0, 1, 5764801, 2239488, 437500, 57344, 5670, 448, 28, 0, 1, 134217728, 51883209, 10077696, 1312500, 129024, 10206, 672, 36, 0, 1

Offset: 0

Author

Alois P. Heinz, Dec 30 2021

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
        1;
        0,       1;
        1,       0,      1;
        8,       3,      0,     1;
       81,      32,      6,     0,    1;
     1024,     405,     80,    10,    0,   1;
    15625,    6144,   1215,   160,   15,   0,  1;
   279936,  109375,  21504,  2835,  280,  21,  0, 1;
  5764801, 2239488, 437500, 57344, 5670, 448, 28, 0, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Column k=0 gives A065440.

Row sums give A204042.

Main diagonal and first lower diagonal give A000012, A000004.

T(n+1,n-1) gives A000217.

T(n+3,n) gives A130809.

T(n+3,n-1) gives A102741 for n>=1.

Cf. A055134, A217701, A350212, A350454.

Programs

Maple

T:= (n, k)-> binomial(n, k)*(n-k-1)^(n-k):
seq(seq(T(n, k), k=0..n), n=0..10);

Formula

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

From Mélika Tebni, Apr 02 2023: (Start)

E.g.f. of column k: -x / (LambertW(-x)*(1+LambertW(-x)))*x^k / k!.

Sum_{k=0..n} k^k*T(n,k) = A217701(n). (End)