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

0,7

Links

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

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)

Example

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;
  ...

Maple

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

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.

Sequence in context: A107671 A271174 A216891 * A256783 A154538 A154166

Adjacent sequences: A349451 A349452 A349453 * A349455 A349456 A349457

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 30 2021

Status

approved