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A246214 Number of endofunctions on [n] where the largest cycle length equals 4.

%I A246214 #9 Dec 28 2020 09:53:11
%S A246214 6,150,3240,72030,1719060,44520840,1252364400,38167414560,
%T A246214 1255558958280,44404434904830,1681726757430720,67953913291104750,
%U A246214 2919509551303952880,132943540577100047760,6397727538671302783680,324511272091351156939200,17306903935107005765263200
%N A246214 Number of endofunctions on [n] where the largest cycle length equals 4.
%H A246214 Alois P. Heinz, <a href="/A246214/b246214.txt">Table of n, a(n) for n = 4..200</a>
%F A246214 a(n) ~ (4*exp(25/12) - 3*exp(11/6)) * n^(n-1). - _Vaclav Kotesovec_, Aug 21 2014
%p A246214 with(combinat):
%p A246214 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A246214       add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
%p A246214       b(n-i*j, i-1), j=0..n/i)))
%p A246214     end:
%p A246214 A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, min(j, k)), j=0..n):
%p A246214 a:= n-> A(n, 4) -A(n, 3):
%p A246214 seq(a(n), n=4..25);
%t A246214 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A246214 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 A246214 A[n_, k_] := Sum[Binomial[n-1, j-1] n^(n-j) b[j, Min[j, k]], {j, 0, n}];
%t A246214 a[n_] := A[n, 4] - A[n, 3];
%t A246214 a /@ Range[4, 25] (* _Jean-François Alcover_, Dec 28 2020, after _Alois P. Heinz_ *)
%Y A246214 Column k=4 of A241981.
%K A246214 nonn
%O A246214 4,1
%A A246214 _Alois P. Heinz_, Aug 19 2014