0,4

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 (deutsch(AT)duke.poly.edu), May 08 2008

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.

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

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.

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

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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.

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

Inverse binomial transform of A002627. - Ross La Haye (rlahaye(AT)new.rr.com), Sep 21 2004

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

Denominator[k=1; NestList[1+1/(k++ #1)&, 1, 12]] - Wouter Meeussen (wouter.meeussen(AT)pandora.be), Mar 24 2007

(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))

Equals n! - A000166(n), i.e. A000142-A000166. Cf. A002468, A002469, A028306, etc.

Row sums of A068106.

Cf. A052169.

Sequence in context: A086365 A032270 A002750 this_sequence A111726 A090376 A125307

Adjacent sequences: A002464 A002465 A002466 this_sequence A002468 A002469 A002470

nonn,easy,nice

njas, Jeffrey Shallit