A246609 Number T(n,k) of endofunctions on [n] whose cycle lengths are multiples of k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

1, 0, 1, 0, 4, 1, 0, 27, 6, 2, 0, 256, 57, 24, 6, 0, 3125, 680, 300, 120, 24, 0, 46656, 9945, 4480, 2160, 720, 120, 0, 823543, 172032, 78750, 41160, 17640, 5040, 720, 0, 16777216, 3438673, 1591296, 866460, 430080, 161280, 40320, 5040

Offset: 0

Author

Alois P. Heinz, Aug 31 2014

Keywords

• nonn
• tabl

Comments

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k <= n. T(0,k) = 1, T(n,k) = 0 for k > n and n > 0.

Column k > 1 is asymptotic to n^(n - 1/2 + 1/(2*k)) * sqrt(2*Pi) / (2^(1/(2*k)) * k^(1/k) * Gamma(1/(2*k))) * (1 - (3*k-1)*(k-1) * sqrt(2/n) * Gamma(1/(2*k)) / (12 * k^2 * Gamma(1/2+1/(2*k)))). - Vaclav Kotesovec, Sep 01 2014

Examples

			Triangle T(n,k) begins:
  1;
  0,      1;
  0,      4,      1;
  0,     27,      6,     2;
  0,    256,     57,    24,     6;
  0,   3125,    680,   300,   120,    24;
  0,  46656,   9945,  4480,  2160,   720,  120;
  0, 823543, 172032, 78750, 41160, 17640, 5040, 720;
  ...
		

Links

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

Crossrefs

Columns k=0-10 give: A000007, A000312, A060435, A246610, A246611, A246612, A246613, A246614, A246615, A246616, A246617.

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

T(2n,n) gives A246618.

Programs

Maple

with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0 or i>n, 0, add(b(n-i*j, i+k, k)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
T:= (n, k)->add(b(j, k$2)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(seq(T(n,k), k=0..n), n=0..10);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0 || i > n, 0, Sum[b[n-i*j, i+k, k]*(i-1)!^j*multinomial[n, {n-i*j, Sequence @@ Table[i, {j}]}]/j!, {j, 0, n/i}]]]; T[0, 0] = 1; T[n_, k_] := Sum[b[j, k, k]*n^(n-j)*Binomial[n-1, j-1], {j, 0, n}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)

Formula

E.g.f. for column k > 0: 1 / (1 - (-1)^k * LambertW(-x)^k)^(1/k). - Vaclav Kotesovec, Sep 01 2014