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A351901 Number of permutations of [n] having at least one repeated cycle length.

%I A351901 #23 Apr 19 2022 06:28:35
%S A351901 0,0,1,1,10,46,246,1926,16080,143424,1397520,16163280,190902240,
%T A351901 2534113440,35501044320,531674569440,8558324490240,147103748144640,
%U A351901 2631981703680000,50393537347829760,1011054905709004800,21229069614652569600,468171587690550374400
%N A351901 Number of permutations of [n] having at least one repeated cycle length.
%H A351901 Alois P. Heinz, <a href="/A351901/b351901.txt">Table of n, a(n) for n = 0..450</a>
%H A351901 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A351901 E.g.f.: 1/(1-x) - Product_{j>=1} (1 + x^j/j).
%F A351901 a(n) = A000142(n) - A007838(n).
%F A351901 Limit_{n-> infinity} a(n)/n! = 1 - exp(-gamma) = A227242 = 0.43854... .
%e A351901 a(2) = 1: (1)(2).
%e A351901 a(3) = 1: (1)(2)(3).
%e A351901 a(4) = 10: (1)(2)(3)(4), (1)(2)(3,4), (1)(2,4)(3), (1)(2,3)(4), (1,4)(2)(3), (1,3)(2)(4), (1,2)(3)(4), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3).
%p A351901 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A351901       b(n, i-1)+b(n-i, min(i-1, n-i))/i))
%p A351901     end:
%p A351901 a:= n-> n!*(1-b(n$2)):
%p A351901 seq(a(n), n=0..23);
%t A351901 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
%t A351901      b[n, i - 1] + b[n - i, Min[i - 1, n - i]]/i]];
%t A351901 a[n_] := n!*(1 - b[n, n]);
%t A351901 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Apr 19 2022, after _Alois P. Heinz_ *)
%Y A351901 Cf. A000142, A001620, A007838, A227242.
%K A351901 nonn
%O A351901 0,5
%A A351901 _Alois P. Heinz_, Feb 24 2022