|
%I #9 Dec 28 2020 09:53:04
%S 2,32,500,8600,165690,3568768,85372280,2251589600,65007768650,
%T 2041482333440,69330316507452,2533173484572640,99124829660524850,
%U 4137148176815360000,183498069976613613680,8620747043700633797888,427712115490907106172050,22350263436559575406220800
%N Number of endofunctions on [n] where the largest cycle length equals 3.
%H Alois P. Heinz, <a href="/A246213/b246213.txt">Table of n, a(n) for n = 3..200</a>
%F a(n) ~ (3*exp(11/6)-2*exp(3/2)) * n^(n-1). - _Vaclav Kotesovec_, Aug 21 2014
%p with(combinat):
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
%p b(n-i*j, i-1), j=0..n/i)))
%p end:
%p A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, min(j, k)), j=0..n):
%p a:= n-> A(n, 3) -A(n, 2):
%p seq(a(n), n=3..25);
%t multinomial[n_, k_List] := n!/Times @@ (k!);
%t b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[(i - 1)!^j multinomial[n, Join[{n - i*j}, Table[i, {j}]]]/j! b[n - i*j, i - 1], {j, 0, n/i}]]];
%t A[n_, k_] := Sum[Binomial[n-1, j-1] n^(n-j) b[j, Min[j, k]], {j, 0, n}];
%t a[n_] := A[n, 3] - A[n, 2];
%t a /@ Range[3, 25] (* _Jean-François Alcover_, Dec 28 2020, after _Alois P. Heinz_ *)
%Y Column k=3 of A241981.
%K nonn
%O 3,1
%A _Alois P. Heinz_, Aug 19 2014
|
