OFFSET
0,4
COMMENTS
Number of permutations p of [n] such that p(k) = k+2 for exactly one k in the range 0 < k < n-1. - Vladeta Jovovic, Nov 30 2007
LINKS
Daniel J. Mundfrom, A problem in permutations: the game of 'Mousetrap', European J. Combin. 15 (1994), no. 6, 555-560.
FORMULA
From Vladeta Jovovic, Nov 30 2007: (Start)
a(n) = (n-2)*A055790(n-2).
E.g.f.: 2*x*exp(-x)/(1-x)^3. (End)
a(n) = floor((n!+1)/e) - floor(((n-2)!+1)/e), n > 2. - Gary Detlefs, Mar 27 2011
G.f.: (1-x)*x/Q(0) - x, where Q(k) = 1 + x - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
G.f.: G(0)*x - x, where G(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - (1-x*(1+2*k))*(1-x*(3+2*k))/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 05 2014
MATHEMATICA
Join[{0, 0}, With[{nn=30}, CoefficientList[Series[(2x Exp[-x])/(1-x)^3, {x, 0, nn}], x] Range[0, nn]!]] (* Harvey P. Dale, Nov 16 2013 *)
PROG
(PARI)
C=binomial;
a(n)=if(n<=2, 0, n! + sum(k=1, n, (-1)^k * ( C(n-1, k)+C(n-2, k-1) )*(n-k)! ) );
/* Joerg Arndt, Apr 22 2013 */
(Sage)
def A():
a, b, n = 1, 1, 1
yield 0
while True:
yield b - a
n += 1
a, b = b, (n-2)*a+(n-1)*b
A018934 = A()
print([next(A018934) for _ in range(24)]) # Peter Luschny, Jan 30 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Nov 30 2007, corrected Jan 25 2008
STATUS
approved