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A241982 Number of endofunctions on [2n] where the largest cycle length equals n.

%I A241982 #21 Aug 17 2017 07:59:07
%S A241982 1,3,93,8600,1719060,604727424,331079253120,260480095349760,
%T A241982 278592031202284800,388855261570122547200,686533182382689959116800,
%U A241982 1495779844806108697677004800,3942052104672989614027181260800,12360865524060039746012601384960000
%N A241982 Number of endofunctions on [2n] where the largest cycle length equals n.
%H A241982 Alois P. Heinz, <a href="/A241982/b241982.txt">Table of n, a(n) for n = 0..198</a>
%F A241982 a(n) = A241981(2n,n).
%F A241982 a(n) ~ 2^(3*n+1/2) * n^(2*n-1) / exp(n). - _Vaclav Kotesovec_, Aug 19 2014
%e A241982 a(1) = 3: (1,1), (1,2), (2,2).
%p A241982 with(combinat):
%p A241982 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A241982       add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
%p A241982       b(n-i*j, i-1), j=0..n/i)))
%p A241982     end:
%p A241982 A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, min(j, k)), j=0..n):
%p A241982 a:= n-> `if`(n=0, 1, A(2*n, n) -A(2*n, n-1)):
%p A241982 seq(a(n), n=0..15);
%t A241982 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A241982 b[n_, i_] := b[n, i] = Which[n==0, 1, i<1, 0, True, 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 A241982 A[n_, k_] := Sum[Binomial[n-1, j-1]*n^(n-j)*b[j, Min[j, k]], {j, 0, n}];
%t A241982 a[n_] := If[n == 0, 1, A[2n, n] - A[2n, n-1]];
%t A241982 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Apr 01 2017, translated from Maple *)
%Y A241982 Cf. A241981, A246050.
%K A241982 nonn
%O A241982 0,2
%A A241982 _Alois P. Heinz_, Aug 10 2014