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A246215 Number of endofunctions on [n] where the largest cycle length equals 5.

%I A246215 #9 Dec 28 2020 09:53:18
%S A246215 24,864,24696,688128,19840464,604727424,19632956112,680195957760,
%T A246215 25130679950376,988325574652416,41277744231187464,1826323584590389248,
%U A246215 85391029667937905184,4209030460729215184896,218223423136426488339744,11875233973816788160610304
%N A246215 Number of endofunctions on [n] where the largest cycle length equals 5.
%H A246215 Alois P. Heinz, <a href="/A246215/b246215.txt">Table of n, a(n) for n = 5..200</a>
%F A246215 a(n) ~ (5*exp(137/60) - 4*exp(25/12)) * n^(n-1). - _Vaclav Kotesovec_, Aug 21 2014
%p A246215 with(combinat):
%p A246215 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A246215       add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
%p A246215       b(n-i*j, i-1), j=0..n/i)))
%p A246215     end:
%p A246215 A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, min(j, k)), j=0..n):
%p A246215 a:= n-> A(n, 5) -A(n, 4):
%p A246215 seq(a(n), n=5..25);
%t A246215 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A246215 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 A246215 A[n_, k_] := Sum[Binomial[n-1, j-1] n^(n-j) b[j, Min[j, k]], {j, 0, n}];
%t A246215 a[n_] := A[n, 5] - A[n, 4];
%t A246215 a /@ Range[5, 25] (* _Jean-François Alcover_, Dec 28 2020, after _Alois P. Heinz_ *)
%Y A246215 Column k=5 of A241981.
%K A246215 nonn
%O A246215 5,1
%A A246215 _Alois P. Heinz_, Aug 19 2014