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Number of endofunctions on [n] where the largest cycle length equals 9.
2

%I #6 Aug 21 2014 04:37:56

%S 40320,4032000,268329600,15328051200,823379356800,43413497487360,

%T 2298646350000000,123818753182924800,6837492347913427200,

%U 388855261570122547200,22836250136299660220160,1386932177757615390720000,87175183432121364413433600,5671938409008942797114572800

%N Number of endofunctions on [n] where the largest cycle length equals 9.

%H Alois P. Heinz, <a href="/A246219/b246219.txt">Table of n, a(n) for n = 9..200</a>

%F a(n) ~ (9*exp(7129/2520) - 8*exp(761/280)) * n^(n-1). - _Vaclav Kotesovec_, Aug 21 2014

%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-> A(n, 9) -A(n, 8):

%p seq(a(n), n=9..25);

%Y Column k=9 of A241981.

%K nonn

%O 9,1

%A _Alois P. Heinz_, Aug 19 2014