A232248 Denominators of the expected length of a random cycle in a random permutation.

1, 2, 12, 24, 720, 1440, 60480, 4480, 3628800, 1036800, 479001600, 958003200, 2615348736000, 172204032, 2414168064000, 62768369664000, 2462451425280000, 9146248151040000, 51090942171709440000, 136216903680000, 33720021833328230400000, 67440043666656460800000

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..250

Formula

a(n) = Denominator( 1/(n-1)! * Sum_{i=1..n} A132393(n,i)/i ). - Alois P. Heinz, Nov 23 2013

a(n) = denominator(Sum_{k=0..n} A002657(k)/A091137(k)) (conjectured). - Michel Marcus, Jul 19 2019

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(add(multinomial(n, n-i*j, i$j)/j!*(i-1)!^j
      *b(n-i*j, i-1) *x^j, j=0..n/i))))
    end:
a:= n->denom((p->add(coeff(p, x, i)/i, i=1..n))(b(n$2))/(n-1)!):
seq(a(n), n=1..30);  # Alois P. Heinz, Nov 21 2013
# second Maple program:
a:= n-> denom(add(abs(combinat[stirling1](n, i))/i, i=1..n)/(n-1)!):
seq(a(n), n=1..30);  # Alois P. Heinz, Nov 23 2013

Mathematica

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}]

Crossrefs

Numerators are A232193.

Sequence in context: A126962 A002207 A181814 * A091137 A347284 A092825

Adjacent sequences: A232245 A232246 A232247 * A232249 A232250 A232251

Keyword

nonn,frac

Author

Geoffrey Critzer, Nov 21 2013

Status

approved