A350454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, none of which are isolated; triangle T(n,k), n >= 0, 0 <= k <= n/2, read by rows.

1, 0, 1, 2, 8, 9, 81, 76, 12, 1024, 875, 180, 15625, 12606, 2910, 120, 279936, 217217, 53550, 3780, 5764801, 4348856, 1118936, 102480, 1680, 134217728, 99111735, 26280072, 2817360, 90720, 3486784401, 2532027610, 686569050, 81864720, 3729600, 30240

Offset

0,4

Links

Alois P. Heinz, Rows n = 0..200, flattened

Formula

E.g.f. column k: exp(W(-x))*(-x - W(-x))^k / ((1 + W(-x))*k!), W(x) the Lambert W-function. - Mélika Tebni, Nov 22 2022

From Mélika Tebni, Dec 22 2022: (Start)

For n > 1, T(n,1) = n*A045531(n-1).

Sum_{k=0..n} (-1)^(n-k)*T(n+k,k) = 2^n.

Sum_{k=0..n} (-1)^(n-k)*T(n+k,k)/(n+k-1) = 1/n, for n > 1. (End)

Example

Triangle T(n,k) begins:
           1;
           0;
           1,          2;
           8,          9;
          81,         76,        12;
        1024,        875,       180;
       15625,      12606,      2910,      120;
      279936,     217217,     53550,     3780;
     5764801,    4348856,   1118936,   102480,    1680;
   134217728,   99111735,  26280072,  2817360,   90720;
  3486784401, 2532027610, 686569050, 81864720, 3729600, 30240;
  ...

Maple

c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
t:= proc(n) option remember; n^(n-1) end:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-i)*
      binomial(n-1, i-1)*(c(i)+`if`(i=1, 0, x*t(i))), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..12);
# second Maple program:
egf := k-> exp(LambertW(-x))*(-x-LambertW(-x))^k/((1+LambertW(-x))*k!):
A350454 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A350454(n, k), k=0..n/2)), n=0..9); # Mélika Tebni, Nov 22 2022

Mathematica

c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
t[n_] := t[n] = n^(n - 1);
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - i]*
     Binomial[n - 1, i - 1]*(c[i] + If[i == 1, 0, x*t[i]]), {i, 1, n}]]];
T[n_] := With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 06 2022, after Alois P. Heinz *)

Crossrefs

Column k=0 gives A065440.

Row sums give |A069856|.

T(2n,n) gives A001813.

Cf. A349454.

Sequence in context: A055678 A301281 A091793 * A109351 A111134 A036898

Adjacent sequences: A350451 A350452 A350453 * A350455 A350456 A350457

Keyword

nonn,tabf

Author

Alois P. Heinz, Dec 31 2021

Status

approved