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Column k=1 of A322384.
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a[ n_] := If[ n < 1, 0, 1 + Total @ Apply[ Max, Map[ Length, First /@ PermutationCycles /@ Drop[ Permutations @ Range @ n, 1], {2}], 1]]; (* Michael Somos, Aug 19 2018 *)
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Ph. Flajolet and A. Odlyzko, <a href="http://algo.inria.fr/flajolet/Publications/publistFlOd90b.htmlpdf">Singularity analysis of generating functions</a>, p. 22.
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Sum the n-permutations having at least 1 cycle of length >= i for all i >= 1. A000142 + A033312 + A066052 + A202364 + ... The summation is precisely that indicated in the title since each permutation whose longest cycle = i is counted i times. - Geoffrey Critzer, Jan 09 2013
E.g.f.: sum(Sum_{k>=0, } (1/(1-x) - exp(sum(Sum_{j=1..k, } x^j/j )) ).
a(n) = f(n, 0, n, n!) where f(l,L, r, n, m) = m*r if r >= l, otherwise sumSum_{k=0..L-1} (f(k, max(lL-k,r), n-1, m/n) for k=0 to l-1) + (n-lL)*f(l,L, r, n-1, m/n)). [- _Thomas Dybdahl Ahle, _, Aug 15 2011]
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