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Denominators of the expected length of a random cycle in a random permutation.
4

%I #19 Jul 19 2019 04:23:54

%S 1,2,12,24,720,1440,60480,4480,3628800,1036800,479001600,958003200,

%T 2615348736000,172204032,2414168064000,62768369664000,

%U 2462451425280000,9146248151040000,51090942171709440000,136216903680000,33720021833328230400000,67440043666656460800000

%N Denominators of the expected length of a random cycle in a random permutation.

%H Alois P. Heinz, <a href="/A232248/b232248.txt">Table of n, a(n) for n = 1..250</a>

%F a(n) = Denominator( 1/(n-1)! * Sum_{i=1..n} A132393(n,i)/i ). - _Alois P. Heinz_, Nov 23 2013

%F a(n) = denominator(Sum_{k=0..n} A002657(k)/A091137(k)) (conjectured). - _Michel Marcus_, Jul 19 2019

%p with(combinat):

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

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

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

%p end:

%p a:= n->denom((p->add(coeff(p, x, i)/i, i=1..n))(b(n$2))/(n-1)!):

%p seq(a(n), n=1..30); # _Alois P. Heinz_, Nov 21 2013

%p # second Maple program:

%p a:= n-> denom(add(abs(combinat[stirling1](n, i))/i, i=1..n)/(n-1)!):

%p seq(a(n), n=1..30); # _Alois P. Heinz_, Nov 23 2013

%t Table[Denominator[Total[Map[Total[#]!/Product[#[[i]],{i,1,Length[#]}]/Apply[Times,Table[Count[#,k]!,{k,1,Max[#]}]]/(Total[#]-1)!/Length[#]&,Partitions[n]]]],{n,1,25}]

%Y Numerators are A232193.

%K nonn,frac

%O 1,2

%A _Geoffrey Critzer_, Nov 21 2013