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A028417 Sum over all n! permutations of n elements of minimum lengths of cycles.

%I A028417 #24 Apr 21 2020 09:13:39
%S A028417 1,3,10,45,236,1505,10914,90601,837304,8610129,96625970,1184891081,
%T A028417 15665288484,223149696601,3394965018886,55123430466945,
%U A028417 948479737691504,17289345305870561,332019600921360594,6713316975465246889,142321908843254560540,3161718732648662557161
%N A028417 Sum over all n! permutations of n elements of minimum lengths of cycles.
%H A028417 Alois P. Heinz, <a href="/A028417/b028417.txt">Table of n, a(n) for n = 1..450</a>
%F A028417 E.g.f.: Sum[k>0, -1+ exp(Sum(j>=k, x^j/j))]. - _Vladeta Jovovic_, Jul 26 2004
%F A028417 a(n) = Sum_{k=1..n} k * A145877(n,k). - _Alois P. Heinz_, Jul 28 2014
%p A028417 b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
%p A028417       b(n-j, min(m,j))*binomial(n-1, j-1), j=1..n))
%p A028417     end:
%p A028417 a:= n-> b(n, infinity):
%p A028417 seq(a(n), n=1..25);  # _Alois P. Heinz_, May 14 2016
%t A028417 Drop[Apply[Plus,Table[nn=25;Range[0,nn]!CoefficientList[Series[Exp[Sum[ x^i/i,{i,n,nn}]]-1,{x,0,nn}],x],{n,1,nn}]],1] (* _Geoffrey Critzer_, Jan 10 2013 *)
%t A028417 b[n_, m_] := b[n, m] = If[n == 0, m, Sum[(j-1)! b[n-j, Min[m, j]]* Binomial[n-1, j-1], {j, n}]];
%t A028417 a[n_] := b[n, Infinity];
%t A028417 Array[a, 25] (* _Jean-François Alcover_, Apr 21 2020, after _Alois P. Heinz_ *)
%Y A028417 Cf. A028418, A046746, A006128.
%Y A028417 Cf. A005225.
%Y A028417 Column k=1 of A322383.
%K A028417 nonn
%O A028417 1,2
%A A028417 Joe Keane (jgk(AT)jgk.org)
%E A028417 More terms from _Vladeta Jovovic_, Sep 19 2002