cp's OEIS Frontend

A365229 Sum over all k of 1/k! times the number of permutations of [n] for which the difference between the longest and the shortest cycle length is k.

%I A365229 #12 Aug 28 2023 11:37:32
%S A365229 1,1,2,6,20,85,382,2219,13624,100293,811914,7594015,74507490,
%T A365229 862987151,10327793088,139175089681,1966790900028,30983071424315,
%U A365229 496696984054286,8925920862110603,162253809011669330,3228438870635420315,65677024568975412036,1448358661756969370985
%N A365229 Sum over all k of 1/k! times the number of permutations of [n] for which the difference between the longest and the shortest cycle length is k.
%C A365229 a(0) = 1 by convention.
%H A365229 Alois P. Heinz, <a href="/A365229/b365229.txt">Table of n, a(n) for n = 0..275</a>
%H A365229 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A365229 a(n) = Sum_{k=0..max(0,n-2)} A364967(n,k)/k!.
%F A365229 a(n) mod 2 = A000035(n) for n>=4.
%p A365229 b:= proc(n, l, m) option remember; `if`(n=0, 1/(m-l)!, add((j-1)!
%p A365229       *b(n-j, min(l, j), max(m, j))*binomial(n-1, j-1), j=1..n))
%p A365229     end:
%p A365229 a:= n-> b(n$2, 0):
%p A365229 seq(a(n), n=0..23);
%Y A365229 Cf. A000035, A000142, A364967.
%K A365229 nonn
%O A365229 0,3
%A A365229 _Alois P. Heinz_, Aug 27 2023