A018934 From the game of Mousetrap.

0, 0, 0, 2, 8, 42, 256, 1810, 14568, 131642, 1320128, 14551074, 174879880, 2276108362, 31894886208, 478775722802, 7664993150696, 130369025763930, 2347604596782208, 44619881467365442, 892659329531868168, 18750556523491299434, 412601744979927877760, 9491630163800726992722

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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.

Crossrefs

Cf. A000166, A002468, A055790.

Programs

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 *)

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

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

For n > 1, a(n) = (n-1)*A000166(n-1) + (n-2)*A000166(n-2). - Kevin Long, Feb 21 2021

Extensions

More terms from Vladeta Jovovic, Nov 30 2007, corrected Jan 25 2008