A350212 Number T(n,k) of endofunctions on [n] with exactly k isolated fixed points; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

1, 0, 1, 3, 0, 1, 17, 9, 0, 1, 169, 68, 18, 0, 1, 2079, 845, 170, 30, 0, 1, 31261, 12474, 2535, 340, 45, 0, 1, 554483, 218827, 43659, 5915, 595, 63, 0, 1, 11336753, 4435864, 875308, 116424, 11830, 952, 84, 0, 1, 262517615, 102030777, 19961388, 2625924, 261954, 21294, 1428, 108, 0, 1

Offset

0,4

Links

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

Formula

Sum_{k=0..n} k * T(n,k) = A055897(n).

Sum_{k=1..n} T(n,k) = A350134(n).

From Mélika Tebni, Nov 24 2022: (Start)

T(n,k) = binomial(n, k)*|A069856(n-k)|.

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

T(n,k) = Sum_{j=0..n} (-1)^(j-k)*binomial(j, k)*binomial(n, j)*(n-j)^(n-j). (End)

Example

T(3,1) = 9: 122, 133, 132, 121, 323, 321, 113, 223, 213.
Triangle T(n,k) begins:
         1;
         0,       1;
         3,       0,      1;
        17,       9,      0,      1;
       169,      68,     18,      0,     1;
      2079,     845,    170,     30,     0,   1;
     31261,   12474,   2535,    340,    45,   0,  1;
    554483,  218827,  43659,   5915,   595,  63,  0, 1;
  11336753, 4435864, 875308, 116424, 11830, 952, 84, 0, 1;
  ...

Maple

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, m) option remember; `if`(n=0, x^m, add(g(i)*
      b(n-i, m+`if`(i=1, 1, 0))*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..10);
# second Maple program:
A350212 := (n, k)-> add((-1)^(j-k)*binomial(j, k)*binomial(n, j)*(n-j)^(n-j), j=0..n):
seq(print(seq(A350212(n, k), k=0..n)), n=0..9); # Mélika Tebni, Nov 24 2022

Mathematica

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[g[i]*
     b[n - i, m + If[i == 1, 1, 0]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)

Crossrefs

Columns k=0-1 give: |A069856|, A348590.

Row sums give A000312.

T(n+1,n-1) gives A045943.

Cf. A001865, A008290, A008291, A055134, A055897, A060281, A086659, A124323, A350134, A349454.

Sequence in context: A279031 A304336 A287315 * A256311 A022695 A278325

Adjacent sequences: A350209 A350210 A350211 * A350213 A350214 A350215

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 19 2021

Status

approved