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

0,5

Comments

T(0,0) = 1 by convention.

Links

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

Formula

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

Example

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;
  ...

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 *)

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.

Sequence in context: A131222 A228334 A114151 * A360177 A241981 A147723

Adjacent sequences: A243095 A243096 A243097 * A243099 A243100 A243101

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 18 2014

Status

approved