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

1, 0, 1, 1, 10, 26, 276, 1212, 14736, 92304, 1285920, 10516320, 166112640, 1680462720, 29753498880, 359124192000, 7053661440000, 98989454592000, 2137497610752000, 34210080898560000, 805846718380032000, 14489879077804032000, 369868281883398144000

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450
• Wikipedia, One hundred prisoners.
• Index entries for sequences related to factorial numbers

Crossrefs

A075829(n) = a(n-1)/gcd(a(n-1), a(n)).

Cf. A000142, A024167, A244009.

Programs

Maple

a := n -> n!*((-1)^n*LerchPhi(-1, 1, n + 1) + 1 - log(2));
seq(simplify(a(n)), n=0..21); # Peter Luschny, Dec 27 2018

Mathematica

f[k_] := (k + 1) (-1)^(k + 1)
t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 22}]    (* A024168 signed *)
(* Clark Kimberling, Dec 30 2011 *)

PARI

x='x+O('x^33); concat([0],Vec(serlaplace((x-log(1+x))/(1-x)))) \\ Joerg Arndt, Dec 27 2018

Formula

From Michael Somos, Oct 29 2002: (Start)

E.g.f.: (log(x+1)-1)/(x-1).

a(n) = a(n-1)+a(n-2)*(n-1)^2, n>=2. (End)

a(0) = 1, a(n) = a(n-1)*n + (-1)^n*(n-1)!. - Daniel Suteu, Feb 06 2017

a(n) = n!*((-1)^n*LerchPhi(-1, 1, n+1) + 1 - log(2)). - Peter Luschny, Dec 27 2018

Limit_{n->oo} a(n)/n! = 1 - log(2) = A244009. - Alois P. Heinz, Jul 08 2022

Extensions

More terms from Michael Somos, Oct 29 2002

a(0)=1 prepended and edited by Alois P. Heinz, Sep 24 2023