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A024168 a(n) = n! * (1 + Sum_{j=1..n} (-1)^j/j).

%I A024168 #40 Sep 25 2023 17:28:52
%S A024168 1,0,1,1,10,26,276,1212,14736,92304,1285920,10516320,166112640,
%T A024168 1680462720,29753498880,359124192000,7053661440000,98989454592000,
%U A024168 2137497610752000,34210080898560000,805846718380032000,14489879077804032000,369868281883398144000
%N A024168 a(n) = n! * (1 + Sum_{j=1..n} (-1)^j/j).
%C A024168 a(n) is the number of permutations of n letters all cycles of which have length <= n/2, a quantity which arises in the solution to the One Hundred Prisoners problem. - Jim Ferry (jferry(AT)alum.mit.edu), Mar 29 2007
%H A024168 Alois P. Heinz, <a href="/A024168/b024168.txt">Table of n, a(n) for n = 0..450</a>
%H A024168 Wikipedia, <a href="http://en.wikipedia.org/wiki/Random_Permutation_Statistics#One_hundred_prisoners">One hundred prisoners</a>.
%H A024168 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A024168 From _Michael Somos_, Oct 29 2002: (Start)
%F A024168 E.g.f.: (log(x+1)-1)/(x-1).
%F A024168 a(n) = a(n-1)+a(n-2)*(n-1)^2, n>=2. (End)
%F A024168 a(0) = 1, a(n) = a(n-1)*n + (-1)^n*(n-1)!. - _Daniel Suteu_, Feb 06 2017
%F A024168 a(n) = n!*((-1)^n*LerchPhi(-1, 1, n+1) + 1 - log(2)). - _Peter Luschny_, Dec 27 2018
%F A024168 Limit_{n->oo} a(n)/n! = 1 - log(2) = A244009. - _Alois P. Heinz_, Jul 08 2022
%p A024168 a := n -> n!*((-1)^n*LerchPhi(-1, 1, n + 1) + 1 - log(2));
%p A024168 seq(simplify(a(n)), n=0..21); # _Peter Luschny_, Dec 27 2018
%t A024168 f[k_] := (k + 1) (-1)^(k + 1)
%t A024168 t[n_] := Table[f[k], {k, 1, n}]
%t A024168 a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t A024168 Table[a[n], {n, 1, 22}]    (* A024168 signed *)
%t A024168 (* _Clark Kimberling_, Dec 30 2011 *)
%o A024168 (PARI) x='x+O('x^33); concat([0],Vec(serlaplace((x-log(1+x))/(1-x)))) \\ _Joerg Arndt_, Dec 27 2018
%Y A024168 A075829(n) = a(n-1)/gcd(a(n-1), a(n)).
%Y A024168 Cf. A000142, A024167, A244009.
%K A024168 nonn
%O A024168 0,5
%A A024168 _Clark Kimberling_
%E A024168 More terms from _Michael Somos_, Oct 29 2002
%E A024168 a(0)=1 prepended and edited by _Alois P. Heinz_, Sep 24 2023