A131622 Number of cycles in all permutations of n elements with distinct cycle lengths.

1, 1, 8, 22, 124, 948, 6138, 50832, 468144, 5165280, 54704880, 695854080, 9016051680, 130427750880, 1994479744320, 32575206343680, 555499414471680, 10284817657927680, 196642556903116800, 3994718386866278400, 84989047758544742400, 1895851170953432985600

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..450

Formula

E.g.f.: Sum(x^n/(n+x^n), n=1..inf) * Product(1+x^n/n, n=1..inf).

Maple

A131622 := proc(n) local su, i ; su := add(x^i/(i+x^i), i=1..n+1) ; for i from 1 to n do su := taylor(su*(1+x^i/i), x=0, n+1) ; od: n!*coeftayl(su, x=0, n) ; end: seq(A131622(n), n=1..30) ; # R. J. Mathar, Oct 25 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, 0, `if`(i>n, 0, (p->[0, p[1]]+p)(
       b(n-i, i-1)*binomial(n, i)*(i-1)!))+b(n, i-1)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..30);  # Alois P. Heinz, May 14 2016

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, If[i > n, {0, 0}, Function[p, {0, p[[1]]} + p][b[n-i, i-1] Binomial[n, i] (i-1)!]] + b[n, i-1]]];
a[n_] := b[n, n][[2]];
Array[a, 30] (* Jean-François Alcover, May 22 2020, after Alois P. Heinz *)
nmax = 30; Rest[CoefficientList[Series[Sum[x^k/(k + x^k), {k, 1, nmax}] * Product[1 + x^k/k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, May 22 2020 *)

Crossrefs

Cf. A000254, A007838.

Sequence in context: A033456 A264631 A026593 * A217275 A183308 A362825

Adjacent sequences: A131619 A131620 A131621 * A131623 A131624 A131625

Keyword

easy,nonn

Author

Vladeta Jovovic, Sep 02 2007

Extensions

More terms from R. J. Mathar, Oct 25 2007

Status

approved