OFFSET
0,5
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.
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
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 *)
PROG
(PARI) x='x+O('x^33); concat([0], Vec(serlaplace((x-log(1+x))/(1-x)))) \\ Joerg Arndt, Dec 27 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Michael Somos, Oct 29 2002
a(0)=1 prepended and edited by Alois P. Heinz, Sep 24 2023
STATUS
approved