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A246191 Number of endofunctions on [n] where the smallest cycle length equals 4.

%I A246191 #9 Dec 28 2020 09:52:50
%S A246191 6,120,2160,41160,861420,19949328,510320160,14348862000,440879024520,
%T A246191 14716697990280,530761366078944,20577610843203960,853717568817968400,
%U A246191 37746072677473752480,1771994498414094109440,88032162789004128733152,4614300279345812506938720
%N A246191 Number of endofunctions on [n] where the smallest cycle length equals 4.
%H A246191 Alois P. Heinz, <a href="/A246191/b246191.txt">Table of n, a(n) for n = 4..200</a>
%F A246191 a(n) ~ (exp(-11/6) - exp(-25/12)) * n^n. - _Vaclav Kotesovec_, Aug 21 2014
%p A246191 with(combinat):
%p A246191 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
%p A246191       add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
%p A246191       b(n-i*j, i+1), j=0..n/i)))
%p A246191     end:
%p A246191 A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, k), j=0..n):
%p A246191 a:= n-> A(n, 4) -A(n, 5):
%p A246191 seq(a(n), n=4..25);
%t A246191 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A246191 b[n_, i_] := b[n, i] = If[n==0, 1, If[i>n, 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 A246191 A[n_, k_] := Sum[Binomial[n - 1, j - 1] n^(n - j) b[j, k], {j, 0, n}];
%t A246191 a[n_] := A[n, 4] - A[n, 5];
%t A246191 a /@ Range[4, 25] (* _Jean-François Alcover_, Dec 28 2020, after _Alois P. Heinz_ *)
%Y A246191 Column k=4 of A246049.
%K A246191 nonn
%O A246191 4,1
%A A246191 _Alois P. Heinz_, Aug 18 2014