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

0,4

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

Table of n, a(n) for n=0..23.

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

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

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

Cf. A000166, A002468, A055790.

Sequence in context: A133417 A235350 A100327 * A107588 A013999 A130649

Adjacent sequences: A018931 A018932 A018933 * A018935 A018936 A018937

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved