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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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72
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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, 32295634916658495460, 710503968166486900119
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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). - Gary W. Adamson, Dec 29 2008
a(n) is 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
Numerator of partial sum of alternating harmonic series, provided that the denominator is n!. - Richard Locke Peterson, May 11 2020
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REFERENCES
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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 Erdős 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).
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LINKS
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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 and R. J. Nowakowski, Mousetrap, Preprint, Feb 10 1993 [Annotated scanned copy]
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-1)*(a(n-1) + a(n-2)), n > 1; a(1) = 1. - Michael Somos, Aug 11 1999
a(0) = 0, a(n) = floor(n!(e-1)/e + 1/2) for n > 0. - Michael Somos, Aug 11 1999
a(n) = - n! * Sum_{i=1..n} (-1)^i/i!. Limit_{n->infinity} a(n)/n! = 1 - 1/e. - Gerald McGarvey, Jun 08 2004
a(n) = (n-1)*(a(n-1) + a(n-2)), n > 1. - Gary Detlefs, Apr 11 2010
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
a(n) = Gamma(n+1) - Gamma(n+1, -1)*exp(-1). - Peter Luschny, Feb 28 2017
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EXAMPLE
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G.f. = x + x^2 + 4*x^3 + 15*x^4 + 76*x^5 + 455*x^6 + 3186*x^7 + 25487*x^8 + ...
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MAPLE
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a := proc(n) -add((-1)^i*binomial(n, i)*(n-i)!, i=1..n) end;
a := n->-n!*add((-1)^k/k!, k=1..n): seq(a(n), n=0..20); # Zerinvary Lajos, May 25 2007
a := n -> simplify(GAMMA(n+1) - GAMMA(n+1, -1)*exp(-1)):
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MATHEMATICA
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Denominator[k=1; NestList[1+1/(k++ #1)&, 1, 12]] (* Wouter Meeussen, Mar 24 2007 *)
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 *)
RecurrenceTable[{a[n] == (n - 1) ( a[n - 1] + a[n - 2]), a[0] == 0, a[1] == 1}, a[n], {n, 20}] (* Ray Chandler, Jul 30 2015 *)
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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