A208248 Sum of the maximum cycle length over all functions f:{1,2,...,n} -> {1,2,...,n} (endofunctions).

0, 1, 5, 40, 431, 5826, 94657, 1795900, 38963535, 951398890, 25819760021, 770959012704, 25117397416795, 886626537549130, 33708625339151505, 1373237757290215156, 59677939242566840303, 2755753623830236494930, 134746033233724391374765, 6954962673986411576581000

Offset: 0

Author

Geoffrey Critzer, Jan 12 2013

Keywords

• nonn

Comments

a(n) is also the sum of the number of endofunctions with at least one cycle >= i for all i >= 1. In other words, a(n) = A000312(n) + A101334(n) + A208240(n) + ... .

Links

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

Crossrefs

Cf. A000169, A028418, A000312, A101334, A208231, A208240, A290932.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
      b(n-j, max(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> add(b(j, 0)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);  # 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[1/(1-t) - Exp[Sum[t^i/i, {i,1,n}]], {x,0,nn}], x], {n, 0, nn-1}]]

Formula

E.g.f.: Sum_{k>=0} 1/(1-T(x)) - exp(Sum_{i=1...k} T(x)^i/i) = A(T(x)) where A(x) is the e.g.f. for A028418 and T(x) is the e.g.f. for A000169.