A350446 Number T(n,k) of endofunctions on [n] with exactly k cycles of length larger than 1; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

1, 1, 3, 1, 16, 11, 125, 128, 3, 1296, 1734, 95, 16807, 27409, 2425, 15, 262144, 499400, 61054, 945, 4782969, 10346328, 1605534, 42280, 105, 100000000, 240722160, 44981292, 1706012, 11025, 2357947691, 6222652233, 1351343346, 67291910, 763875, 945

Offset: 0

Author

Alois P. Heinz, Dec 31 2021

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
           1;
           1;
           3,          1;
          16,         11;
         125,        128,          3;
        1296,       1734,         95;
       16807,      27409,       2425,       15;
      262144,     499400,      61054,      945;
     4782969,   10346328,    1605534,    42280,    105;
   100000000,  240722160,   44981292,  1706012,  11025;
  2357947691, 6222652233, 1351343346, 67291910, 763875, 945;
  ...
		

Links

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

Crossrefs

Column k=0 gives A000272(n+1).

Row sums give A000312.

T(2n,n) gives A001147.

Cf. A055134, A060281, A349454, A350452.

Cf. A136394, A190314, A217701.

Programs

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)*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-> (LambertW(-x)-log(1+LambertW(-x)))^k/(exp(LambertW(-x))*k!):
A350446 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A350446(n, k), k=0..n/2)), n=0..10); # Mélika Tebni, Mar 23 2023

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]*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 *)

Formula

From Mélika Tebni, Mar 23 2023: (Start)

E.g.f. of column k: (W(-x)-log(1 + W(-x)))^k / (exp(W(-x))*k!), W(x) the Lambert W-function.

T(n,k) = Sum_{j=k..n} n^(n-j)*binomial(n-1,j-1)*A136394(j,k), for n > 0.

T(n,k) = Sum_{j=k..n} (n-j+1)^(n-j-1)*binomial(n,j)*A350452(j,k).

Sum_{k=0..n/2} (k+1)*T(n,k) = A190314(n), for n > 0.

Sum_{k=0..n/2} 2^k*T(n,k) = A217701(n). (End)