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A246190 Number of endofunctions on [n] where the smallest cycle length equals 3.

%I A246190 #11 Dec 28 2020 09:52:41
%S A246190 2,24,300,4360,74130,1456224,32562152,817596000,22785399450,
%T A246190 697951656160,23306666102148,842567564800416,32781106696806650,
%U A246190 1365579024023558400,60639189588419033040,2859165143013913590016,142651621238828972159538,7508140027468431374563200
%N A246190 Number of endofunctions on [n] where the smallest cycle length equals 3.
%H A246190 Alois P. Heinz, <a href="/A246190/b246190.txt">Table of n, a(n) for n = 3..200</a>
%F A246190 a(n) ~ (exp(-3/2) - exp(-11/6)) * n^n. - _Vaclav Kotesovec_, Aug 21 2014
%p A246190 with(combinat):
%p A246190 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
%p A246190       add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
%p A246190       b(n-i*j, i+1), j=0..n/i)))
%p A246190     end:
%p A246190 A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, k), j=0..n):
%p A246190 a:= n-> A(n, 3) -A(n, 4):
%p A246190 seq(a(n), n=3..25);
%t A246190 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A246190 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 A246190 A[n_, k_] := Sum[Binomial[n - 1, j - 1] n^(n - j) b[j, k], {j, 0, n}];
%t A246190 a[n_] := A[n, 3] - A[n, 4];
%t A246190 a /@ Range[3, 25] (* _Jean-François Alcover_, Dec 28 2020, after _Alois P. Heinz_ *)
%Y A246190 Column k=3 of A246049.
%K A246190 nonn
%O A246190 3,1
%A A246190 _Alois P. Heinz_, Aug 18 2014