A246531 Number of endofunctions on [n] whose cycle lengths are divisors of n.

1, 1, 4, 18, 224, 1320, 42552, 262864, 12232320, 169594560, 6117023600, 61920993024, 8022787347456, 56694391376896, 5193025319432160, 174746314698336000, 10338252997184749568, 121439552019384139776, 26096843176349347142208, 262144006402373705728000

Offset: 0

Author

Alois P. Heinz, Aug 28 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Main diagonal of A246522.

Programs

Maple

with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(n), x, n+1), x, n):
seq(a(n), n=0..20);
# second Maple program:
with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1, k)*
      (i-1)!^j, j=0..`if`(irem(k, i)=0, n/i, 0))))
    end:
a:= n-> add(b(j$2, n)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
     Sum[multinomial[n, Join[{n - i*j},
     Table[i, {j}]]]/j!*b[n - i*j, i - 1, k]*(i - 1)!^j,
     {j, 0, If[Mod[k, i] == 0, n/i, 0]}]]];
a[n_] := If[n==0, 1, Sum[b[j, j, n]*n^(n-j)*Binomial[n-1, j-1], {j, 0, n}]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 01 2022, after Alois P. Heinz *)

Formula

a(n) = n! * [x^n] exp(Sum_{d|n} (-LambertW(-x))^d/d).

a(n) = A246522(n,n).