A291116 Number of endofunctions on [n] such that the LCM of their cycle lengths equals ten.

0, 0, 0, 0, 0, 0, 0, 504, 32256, 1460592, 59814720, 2403157680, 98055619200, 4129943329512, 180976836968928, 8281570545448200, 396324506640142080, 19840151844921504096, 1038497761573246945152, 56790713866712335971552, 3241264004352759793685760

Offset: 0

Author

Alois P. Heinz, Aug 17 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..387

Crossrefs

Column k=10 of A222029.

Programs

Maple

b:= proc(n, m) option remember; (k-> `if`(m>k, 0,
      `if`(n=0, `if`(m=k, 1, 0), add(b(n-j, ilcm(m, j))
       *binomial(n-1, j-1)*(j-1)!, j=1..n))))(10)
    end:
a:= n-> add(b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..22);

Mathematica

Unprotect[Power]; Power[0|0., 0|0.]=1; Protect[Power];b[n_, m_]:=b[n, m]=If[m>10, 0, If[n==0, If[m==10,1, 0], Sum[b[n - j, LCM[m, j]] Binomial[n - 1, j - 1](j - 1)!, {j, n}]]]; Table[Sum[b[j, 1]*n^(n -j) Binomial[n - 1, j - 1], {j, 0, n}], {n, 0, 25}] (* Indranil Ghosh, Aug 18 2017 *)

Python

from sympy.core.cache import cacheit
from sympy import binomial, lcm, factorial as f
@cacheit
def b(n, m): return 0 if m>10 else (1 if m==10 else 0) if n==0 else sum([b(n - j, lcm(m, j))*binomial(n - 1, j - 1)*f(j - 1) for j in range(1, n + 1)])
def a(n): return sum([b(j, 1)*n**(n - j)*binomial(n - 1, j - 1) for j in range(n + 1)])
print([a(n) for n in range(26)]) # Indranil Ghosh, Aug 18 2017

Formula

a(n) ~ (exp(1) - 2*exp(3/2) - 2*exp(6/5) + 4*exp(9/5)) * n^(n-1). - Vaclav Kotesovec, Aug 18 2017