A208231 Sum of the minimum cycle length over all functions f:{1,2,...,n}->{1,2,...,n} (endofunctions).

0, 1, 5, 37, 373, 4761, 73601, 1336609, 27888281, 657386305, 17276807089, 500876786301, 15879053677697, 546470462226313, 20288935994319929, 808320431258439121, 34397370632215764001, 1557106493482564625793, 74713970491718324746529, 3787792171563440619543133, 202314171910557294992453009

Offset: 0

Author

Geoffrey Critzer, Jan 10 2013

Keywords

• nonn

Comments

Sum of the number of endofunctions whose cycle lengths are >=i for all i >=1. A000312 + A065440 + A134362 + A208230 + ...

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386

Crossrefs

Cf. A000169, A028417, A000312, A065440, A134362, A208230, A208248, A290932.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
      b(n-j, min(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> add(b(j$2)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, May 20 2016

Mathematica

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Apply[Plus,Table[Range[0,nn]!CoefficientList[Series[Exp[Sum[t^i/i,{i,n,nn}]]-1,{x,0,nn}],x],{n,1,nn}]]

Formula

E.g.f.: A(T(x)) = Sum_{k>=1} exp( Sum_{i>=k} T(x)^i/i) - 1 where A(x) is the e.g.f. for A028417 and T(x) is the e.g.f. for A000169.