A246531 - OEIS

%I #9 May 01 2022 14:01:24

%S 1,1,4,18,224,1320,42552,262864,12232320,169594560,6117023600,

%T 61920993024,8022787347456,56694391376896,5193025319432160,

%U 174746314698336000,10338252997184749568,121439552019384139776,26096843176349347142208,262144006402373705728000

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

%H Alois P. Heinz, <a href="/A246531/b246531.txt">Table of n, a(n) for n = 0..200</a>

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

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

%p with(numtheory):

%p egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):

%p a:= n-> n!*coeff(series(egf(n), x, n+1), x, n):

%p seq(a(n), n=0..20);

%p # second Maple program:

%p with(combinat):

%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1, k)*

%p       (i-1)!^j, j=0..`if`(irem(k, i)=0, n/i, 0))))

%p     end:

%p a:= n-> add(b(j$2, n)*n^(n-j)*binomial(n-1, j-1), j=0..n):

%p seq(a(n), n=0..20);

%t multinomial[n_, k_List] := n!/Times @@ (k!);

%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,

%t      Sum[multinomial[n, Join[{n - i*j},

%t      Table[i, {j}]]]/j!*b[n - i*j, i - 1, k]*(i - 1)!^j,

%t      {j, 0, If[Mod[k, i] == 0, n/i, 0]}]]];

%t a[n_] := If[n==0, 1, Sum[b[j, j, n]*n^(n-j)*Binomial[n-1, j-1], {j, 0, n}]];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 01 2022, after _Alois P. Heinz_ *)

%Y Main diagonal of A246522.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 28 2014