A243098 Number T(n,k) of endofunctions on [n] with all cycles of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 3, 1, 0, 16, 6, 2, 0, 125, 51, 24, 6, 0, 1296, 560, 300, 120, 24, 0, 16807, 7575, 4360, 2160, 720, 120, 0, 262144, 122052, 73710, 41160, 17640, 5040, 720, 0, 4782969, 2285353, 1430016, 861420, 430080, 161280, 40320, 5040

Offset: 0

Author

Alois P. Heinz, Aug 18 2014

Keywords

• nonn
• tabl

Comments

T(0,0) = 1 by convention.

Examples

			Triangle T(n,k) begins:
  1;
  0,      1;
  0,      3,      1;
  0,     16,      6,     2;
  0,    125,     51,    24,     6;
  0,   1296,    560,   300,   120,    24;
  0,  16807,   7575,  4360,  2160,   720,  120;
  0, 262144, 122052, 73710, 41160, 17640, 5040, 720;
  ...
		

Links

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

Crossrefs

Columns k=0-4 give: A000007, A000272(n+1) for n>0, A057817(n+1), 2*A060917, 6*A060918.

Row sums give A241980.

T(2n,n) gives A246050.

Main diagonal gives A000142(n-1) for n>0.

Cf. A241981, A246049.

Programs

Maple

with(combinat):
T:= (n, k)-> `if`(k*n=0, `if`(k+n=0, 1, 0),
    add(binomial(n-1, j*k-1)*n^(n-j*k)*(k-1)!^j*
    multinomial(j*k, k$j, 0)/j!, j=0..n/k)):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

multinomial[n_, k_] := n!/Times @@ (k!); T[n_, k_] := If[k*n==0, If[k+n == 0, 1, 0], Sum[Binomial[n-1, j*k-1]*n^(n-j*k)*(k-1)!^j*multinomial[j*k, Append[Array[k&, j], 0]]/j!, {j, 0, n/k}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

Formula

E.g.f. of column k>0: exp((-LambertW(-x))^k/k), e.g.f. of column k=0: 1.