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A002467 The game of Mousetrap with n cards (given n letters and n envelopes, how many ways are there to fill the envelopes so that at least one letter goes into its right envelope?).
(Formerly M3507 N1423)
37
0, 1, 1, 4, 15, 76, 455, 3186, 25487, 229384, 2293839, 25232230, 302786759, 3936227868, 55107190151, 826607852266, 13225725636255, 224837335816336, 4047072044694047, 76894368849186894, 1537887376983737879 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n) is the number of permutations in the symmetric group S_n that have a fixed point, i.e., they are not derangements (A000166). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001

a(n+1)=p(n+1) where p(x) is the unique degree-n polynomial such that p(k)=k! for k=0,1,...,n. - Michael Somos, Oct 07 2003

The termwise sum of this sequence and A000166 gives the factorial numbers. - D. G. Rogers, Aug 26 2006, Jan 06 2008

a(n) is the number of deco polyominoes of height n and having in the last column an odd number of cells. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=1 because the horizontal domino is the only deco polyomino of height 2 having an odd number of cells in the last column. - Emeric Deutsch, May 08 2008

Starting (1, 4, 15, 76, 455,...) = eigensequence of triangle A127899 (unsigned). - Gary W. Adamson, Dec 29 2008

(n-1) | a(n), hence a(n) is never prime. - Jonathan Vos Post, Mar 25 2009

a(n) = (n-1)*(a(n-1) + a(n-2)), n>1. - Gary Detlefs, Apr 11 2010

a(n) = the number of permutations of [n] that have at least one fixed point = number of positive terms in n-th row of the triangle in A170942, n>0. - Reinhard Zumkeller, Mar 29 2012

REFERENCES

R. K. Guy, Unsolved Problems Number Theory, E37.

R. K. Guy and R. J. Nowakowski, "Mousetrap," in D. Miklos, V.T. Sos and T. Szonyi, eds., Combinatorics, Paul Erdos is Eighty. Bolyai Society Math. Studies, Vol. 1, pp. 193-206, 1993.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

P. R. de Montmort, On the Game of Thirteen (1713), reprinted in Annotated Readings in the History of Statistics, ed. H. A. David and A. W. F. Edwards, Springer-Verlag, 2001, pp. 25-29.

T. Kotek, J. A. Makowsky, Recurrence Relations for Graph Polynomials on Bi-iterative Families of Graphs, arXiv preprint arXiv:1309.4020, 2013.

Daniel J. Mundfrom, A problem in permutations: the game of `Mousetrap'. European J. Combin. 15 (1994), no. 6, 555-560.

Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7

Simon Plouffe, Exact formulas for Integer Sequences.

A. Steen, Some formulae respecting the game of mousetrap, Quart. J. Pure Applied Math., 15 (1878), 230-241.

L. Takacs, The Problem of Coincidences, Archive for History of Exact Sciences, Volume 21, No. 3, Sept. 1980. pp 229-244, paragraphs 5 and 7.

Eric Weisstein's World of Mathematics, Mousetrap.

FORMULA

a(n) = n! - A000166(n) = A000142(n) - A000166(n).

E.g.f.: (1 - exp(-x)) / (1 - x). - Michael Somos, Aug 11 1999

a(n) = (n-1)*(a(n-1) + a(n-2)), n>1; a(1) = 1. - Michael Somos, Aug 11 1999

a(n) = n*a(n-1) - (-1)^n. - Michael Somos, Aug 11 1999

a(0) = 0, a(n) = [ n!(e-1)/e + 1/2 ] for n > 0. - Michael Somos, Aug 11 1999

a(0) = 0, a(n) = n! * Sum i=1..n (-1)^(n-1)/i! for n > 0. lim n->inf a(n)/n! = 1 - 1/e. - Gerald McGarvey, Jun 08 2004

Inverse binomial transform of A002627. - Ross La Haye, Sep 21 2004

a(n)=n! - floor((n!+1)/e), n>0. - Gary Detlefs, Apr 11 2010

For n>0, a(n) = {(1-1/exp(1))*n!}, where {x} is the nearest integer. - Simon Plouffe, conjectured March 1993, added Feb 17 2011

0 = a(n) * (a(n+1) + a(n+2) - a(n+3)) + a(n+1) * (a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2) * (a(n+2)) if n>=0. - Michael Somos, Jan 25 2014

EXAMPLE

G.f. = x + x^2 + 4*x^3 + 15*x^4 + 76*x^5 + 455*x^6 + 3186*x^7 + 25487*x^8 + ...

MAPLE

a := proc(n) local i; add( (-1)^(i+1)*binomial(n+1, i)*(n+1-i)!, i=1..n+1); end;

a:=n->-n!*sum((-1)^k/k!, k=1..n): seq(a(n), n=0..20); # Zerinvary Lajos, May 25 2007

MATHEMATICA

Denominator[k=1; NestList[1+1/(k++ #1)&, 1, 12]] (* Wouter Meeussen, Mar 24 2007 *)

LinearRecurrence[{#1, #1}, {0, 1}, 21] (* Robert G. Wilson v, Jun 15 2013 *)

a[ n_] := If[ n < 0, 0, n! - Subfactorial[n]] (* Michael Somos, Jan 25 2014 *)

a[ n_] := If[ n < 1, 0, n! - Round[ n! / E]] (* Michael Somos, Jan 25 2014 *)

a[ n_] := If[ n < 0, 0, n! - (-1)^n HypergeometricPFQ[ {- n, 1}, {}, 1]](* Michael Somos, Jan 25 2014 *)

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (1 - Exp[ -x] ) / (1 - x), {x, 0, n}]] (* Michael Somos, Jan 25 2014 *)

PROG

(PARI) {a(n) = if( n<1, 0, n * a(n-1) - (-1)^n)} /* Michael Somos, Mar 24 2003 */

(PARI) {a(n) = if( n<0, 0, n! * polcoeff( (1 - exp( -x + x * O(x^n))) / (1 - x), n))} /* Michael Somos, Mar 24 2003 */

(PARI) a(n)=if(n<1, 0, subst(polinterpolate(vector(n, k, (k-1)!)), x, n+1))

(PARI) A002467(n) = if(n<1, 0, n*A002467(n-1)-(-1)^n); \\ Joerg Arndt, Apr 22 2013

CROSSREFS

Cf. A002468, A002469, A028306, etc.

Row sums of A068106.

Cf. A052169.

Cf. A127899. - Gary W. Adamson, Dec 29 2008

Sequence in context: A198057 A002750 A178887 * A243327 A179511 A111726

Adjacent sequences:  A002464 A002465 A002466 * A002468 A002469 A002470

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane, Jeffrey Shallit

STATUS

approved

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Last modified December 18 09:22 EST 2014. Contains 252122 sequences.