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A246050 Number of endofunctions on [2n] where the smallest cycle length equals n.

%I A246050 #12 Feb 11 2015 10:43:23
%S A246050 1,3,51,4360,861420,302472576,165549605760,130241382036480,
%T A246050 139296260790086400,194427690066299289600,343266609438110040883200,
%U A246050 747889929370001008617062400,1971026055567996899374212710400,6180432763819774878006029844480000
%N A246050 Number of endofunctions on [2n] where the smallest cycle length equals n.
%H A246050 Alois P. Heinz, <a href="/A246050/b246050.txt">Table of n, a(n) for n = 0..190</a>
%F A246050 a(n) = A246049(2n,n) = A243098(2n,n).
%F A246050 a(n) ~ 2^(3*n-1/2) * n^(2*n-1) / exp(n). - _Vaclav Kotesovec_, Aug 19 2014
%p A246050 with(combinat):
%p A246050 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
%p A246050       add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
%p A246050       b(n-i*j, i+1), j=0..n/i)))
%p A246050     end:
%p A246050 A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, k), j=0..n):
%p A246050 a:= n-> `if`(n=0, 1, A(2*n, n) -A(2*n, n+1)):
%p A246050 seq(a(n), n=0..15);
%t A246050 multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, Sum[(i-1)!^j*multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i+1], {j, 0, n/i}]]]; A[n_, k_] := Sum[Binomial[n-1, j-1]*n^(n-j)*b[j, k], {j, 0, n}]; a[n_] := If[n == 0, 1, A[2*n, n] - A[2*n, n+1]]; Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Feb 11 2015, after _Alois P. Heinz_ *)
%Y A246050 Cf. A241982, A246049, A243098.
%K A246050 nonn
%O A246050 0,2
%A A246050 _Alois P. Heinz_, Aug 11 2014