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A002467
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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)
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35
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0, 1, 1, 4, 15, 76, 455, 3186, 25487, 229384, 2293839, 25232230, 302786759, 3936227868, 55107190151, 826607852266, 13225725636255, 224837335816336, 4047072044694047, 76894368849186894, 1537887376983737879
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OFFSET
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0,4
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COMMENTS
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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). [From Gary W. Adamson, Dec 29 2008]
(n-1) | a(n), hence a(n) is never prime. [From Jonathan Vos Post, Mar 25 2009]
a(n) = (n-1)*(a(n-1) + a(n-2)), n>1 [From 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]
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REFERENCES
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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.
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.
D. J. Mundfrom, A problem in permutations: the game of `Mousetrap'. European J. Combin. 15 (1994), no. 6, 555-560.
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).
A. Steen, Some formulae respecting the game of mousetrap, Quart. J. Pure Applied Math., 15 (1878), 230-241.
E. Barcucci, A. del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
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.
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.
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FORMULA
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a(n) = n! - A000166(n) = A000142(n) - A000166(n).
E.g.f.: (1-exp(-x))/(1-x).
a(n) = (n-1)(a(n-1)+a(n-2)), n>1; a(1) = 1.
a(n) = n*a(n-1)-(-1)^n;
a(0) = 0, a(n) = [ n!(e-1)/e + 1/2 ] for n > 0.
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.
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MAPLE
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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 (zerinvarylajos(AT)yahoo.com), May 25 2007
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MATHEMATICA
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Denominator[k=1; NestList[1+1/(k++ #1)&, 1, 12]] - Wouter Meeussen, Mar 24 2007
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PROG
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(PARI) a(n)=if(n<1, 0, n*a(n-1)-(-1)^n)
(PARI) a(n)=if(n<0, 0, n!*polcoeff((1-exp(-x+x*O(x^n)))/(1-x), n))
(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
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CROSSREFS
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Cf. A002468, A002469, A028306, etc.
Row sums of A068106.
Cf. A052169.
Cf. A127899 [From Gary W. Adamson, Dec 29 2008]
Sequence in context: A198057 A002750 A178887 * A179511 A111726 A090376
Adjacent sequences: A002464 A002465 A002466 * A002468 A002469 A002470
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, Jeffrey Shallit
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STATUS
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approved
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