A241982 - OEIS

%I #21 Aug 17 2017 07:59:07

%S 1,3,93,8600,1719060,604727424,331079253120,260480095349760,

%T 278592031202284800,388855261570122547200,686533182382689959116800,

%U 1495779844806108697677004800,3942052104672989614027181260800,12360865524060039746012601384960000

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

%H Alois P. Heinz, <a href="/A241982/b241982.txt">Table of n, a(n) for n = 0..198</a>

%F a(n) = A241981(2n,n).

%F a(n) ~ 2^(3*n+1/2) * n^(2*n-1) / exp(n). - _Vaclav Kotesovec_, Aug 19 2014

%e a(1) = 3: (1,1), (1,2), (2,2).

%p with(combinat):

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*

%p       b(n-i*j, i-1), j=0..n/i)))

%p     end:

%p A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, min(j, k)), j=0..n):

%p a:= n-> `if`(n=0, 1, A(2*n, n) -A(2*n, n-1)):

%p seq(a(n), n=0..15);

%t multinomial[n_, k_List] := n!/Times @@ (k!);

%t 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 A[n_, k_] := Sum[Binomial[n-1, j-1]*n^(n-j)*b[j, Min[j, k]], {j, 0, n}];

%t a[n_] := If[n == 0, 1, A[2n, n] - A[2n, n-1]];

%t Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Apr 01 2017, translated from Maple *)

%Y Cf. A241981, A246050.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Aug 10 2014