Search: id:A002467
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%I A002467 M3507 N1423
%S A002467 0,1,1,4,15,76,455,3186,25487,229384,2293839,25232230,302786759,
%T A002467 3936227868,55107190151,826607852266,13225725636255,224837335816336,
%U A002467 4047072044694047,76894368849186894,1537887376983737879
%N 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?).
%C A002467 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
%C A002467 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
%C A002467 The termwise sum of this sequence and A000166 gives the factorial numbers 
               - D. G. Rogers, Aug 26 2006, Jan 06 2008
%C A002467 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 (deutsch(AT)duke.poly.edu), 
               May 08 2008
%C A002467 Starting (1, 4, 15, 76, 455,...) = eigensequence of triangle A127899 
               (unsigned). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 29 
               2008]
%C A002467 (n-1) | a(n), hence a(n) is never prime. [From Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Mar 25 2009]
%D A002467 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002467 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002467 R. K. Guy, Unsolved Problems Number Theory, E37.
%D A002467 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 A002467 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.
%D A002467 D. J. Mundfrom, A problem in permutations: the game of `Mousetrap'. European 
               J. Combin. 15 (1994), no. 6, 555-560.
%D A002467 A. Steen, Some formulae respecting the game of mousetrap, Quart. J. Pure 
               Applied Math., 15 (1878), 230-241.
%D A002467 E. Barcucci, A. del Lungo and R. Pinzani, "Deco" polyominoes, permutations 
               and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%H A002467 T. D. Noe, <a href="b002467.txt">Table of n, a(n) for n=0..100</a>
%H A002467 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</
               a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A002467 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Mousetrap.html">Link to a section of The World of Mathematics.</a>
%F A002467 E.g.f.: (1-e^(-x))/(1-x). a(n)=(n-1)(a(n-1)+a(n-2)), n>1; or a(1)=1, 
               a(n)=n*a(n-1)-(-1)^n; or a(0)= 0, a(n) = [ n!(e-1)/e + 1/2 ] for 
               n > 0.
%F A002467 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 (Gerald.McGarvey(AT)comcast.net), 
               Jun 08 2004
%F A002467 Inverse binomial transform of A002627. - Ross La Haye (rlahaye(AT)new.rr.com), 
               Sep 21 2004
%p A002467 a := proc(n) local i; add( (-1)^(i+1)*binomial(n+1,i)*(n+1-i)!, i=1..n+1); 
               end;
%p A002467 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
%t A002467 Denominator[k=1; NestList[1+1/(k++ #1)&,1,12]] - Wouter Meeussen (wouter.meeussen(AT)pandora.be), 
               Mar 24 2007
%o A002467 (PARI) a(n)=if(n<1,0,n*a(n-1)-(-1)^n)
%o A002467 (PARI) a(n)=if(n<0,0,n!*polcoeff((1-exp(-x+x*O(x^n)))/(1-x),n))
%o A002467 (PARI) a(n)=if(n<1,0,subst(polinterpolate(vector(n,k,(k-1)!)),x,n+1))
%Y A002467 Equals n! - A000166(n), i.e. A000142-A000166. Cf. A002468, A002469, A028306, 
               etc.
%Y A002467 Row sums of A068106.
%Y A002467 Cf. A052169.
%Y A002467 A127899 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 29 2008]
%Y A002467 Sequence in context: A086365 A032270 A002750 this_sequence A111726 A090376 
               A125307
%Y A002467 Adjacent sequences: A002464 A002465 A002466 this_sequence A002468 A002469 
               A002470
%K A002467 nonn,easy,nice
%O A002467 0,4
%A A002467 N. J. A. Sloane (njas(AT)research.att.com), Jeffrey Shallit

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