A350452 Number T(n,k) of endofunctions on [n] with exactly k connected components and no fixed points; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

1, 0, 0, 1, 0, 8, 0, 78, 3, 0, 944, 80, 0, 13800, 1810, 15, 0, 237432, 41664, 840, 0, 4708144, 1022252, 34300, 105, 0, 105822432, 27098784, 1286432, 10080, 0, 2660215680, 778128336, 47790540, 648900, 945, 0, 73983185000, 24165049920, 1815578160, 36048320, 138600

Offset: 0

Author

Alois P. Heinz, Dec 31 2021

Keywords

• nonn
• tabf

Comments

For k >= 2 and p prime, T(p,k) == 0 (mod 4*p*(p-1)). - Mélika Tebni, Jan 20 2023

Examples

			Triangle T(n,k) begins:
  1;
  0;
  0,          1;
  0,          8;
  0,         78,         3;
  0,        944,        80;
  0,      13800,      1810,       15;
  0,     237432,     41664,      840;
  0,    4708144,   1022252,    34300,    105;
  0,  105822432,  27098784,  1286432,  10080;
  0, 2660215680, 778128336, 47790540, 648900, 945;
  ...
		

Links

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

Crossrefs

Columns k=0-1 give: A000007, A000435.

Row sums give A065440.

T(2n,n) gives A001147.

Cf. A060281, A061356, A350446.

Cf. A071720, A007830, A106828.

Programs

Maple

c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
      b(n-i)*binomial(n-1, i-1)*x*c(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);

Mathematica

c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[
     b[n - i]*Binomial[n - 1, i - 1]*x*c[i], {i, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n/2}]][b[n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)

PARI

\\ here AS1(n,k) gives associated Stirling numbers of 1st kind.
AS1(n,k)={(-1)^(n+k)*sum(i=0, k, (-1)^i * binomial(n, i) * stirling(n-i, k-i, 1) )}
T(n,k) = {if(n==0, k==0, sum(j=k, n, n^(n-j)*binomial(n-1, j-1)*AS1(j,k)))} \\ Andrew Howroyd, Jan 20 2023

Formula

From Mélika Tebni, Jan 20 2023: (Start)

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

-Sum_{k=1..n/2} (-1)^k*T(n,k) = A071720(n+1), for n > 0.

-Sum_{k=1..n/2} (-1)^k*T(n,k) / (n-1) = A007830(n-2), for n > 1.

T(n,k) = Sum_{j=k..n} n^(n-j)*binomial(n-1, j-1)*A106828(j, k) for n > 0. (End)