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%I #24 Jan 21 2023 03:03:30
%S 0,1,5,40,431,5826,94657,1795900,38963535,951398890,25819760021,
%T 770959012704,25117397416795,886626537549130,33708625339151505,
%U 1373237757290215156,59677939242566840303,2755753623830236494930,134746033233724391374765,6954962673986411576581000
%N Sum of the maximum cycle length over all functions f:{1,2,...,n} -> {1,2,...,n} (endofunctions).
%C 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) + ... .
%H Alois P. Heinz, <a href="/A208248/b208248.txt">Table of n, a(n) for n = 0..386</a>
%F 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.
%p b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
%p b(n-j, max(m, j))*binomial(n-1, j-1), j=1..n))
%p end:
%p a:= n-> add(b(j, 0)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, May 20 2016
%t 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}]]
%Y Cf. A000169, A028418, A000312, A101334, A208231, A208240, A290932.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Jan 12 2013
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