0,2

The number of one-to-one sequences that can be formed from n distinct objects.

Related to number of operations of addition and multiplication to evaluate a determinant of order n by cofactor expansion - see A026243.

a(n) is also the number of paths (without loops) in the complete graph on n+2 vertices starting at one vertex v1 and ending at another v2. Example: when n=2 there are 5 paths in the complete graph with 4 vertices starting at the vertex 1 and ending at the vertex 2: (12),(132),(142),(1342),(1432) so a(2) = 5. - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001; comment corrected by Jonathan Coxhead, Mar 21, 2003

Also row sums of Table A008279, which can be generated by taking the derivatives of x^k. For example, for y = 1*x^3, y' = 3x^2, y" = 6x, y'''= 6 so a(4) = 1 + 3 + 6 + 6 = 16 - Alford Arnold (Alford1940), Dec 15 1999

a(n) is the permanent of the n X n matrix with 2s on the diagonal and 1s elsewhere. - Yuval Dekel, Nov 01 2003

(A000166 + this_sequence)/2 = A009179, (A000166 - this_sequence)/2 = A009628.

Stirling transform of A006252(n-1)=[1,1,1,2,4,14,38,...] is a(n-1)=[1,2,5,16,65,...]. - Michael Somos Mar 04 2004

Number of {12,12*,21*}- and {12,12*,2*1}-avoiding signed permutations in the hyperoctahedral group.

a(n)=exp(1)*Gamma(n+1,1) where Gamma(z,t)=Integral_{x>=t} exp(-x)x^(z-1) dx is incomplete gamma function. - Michael Somos Jul 1 2004

a(n) = b such that Integral_{x=0..1} x^n*exp(-x) dx = a-b*exp(-1). - Sebastien DUMORTIER (sdumortier(AT)ac-limoges.fr), Mar 05 2005

a(n) = number of permutations on [n] whose left-to-right record lows all occur at the start. Example: a(3) counts all permutations on [3] except 231 (the last entry is a record low but its predecessor is not). - David Callan (callan(AT)stat.wisc.edu), Jul 20 2005

a(n) = number of permutations on [n+1] that avoid the (scattered) pattern 1-2-3|. The vertical bar means the "3" must occur at the end of the permutation. For example, 21354 is not counted by a(4): 234 is an offending subpermutation. - David Callan (callan(AT)stat.wisc.edu), Nov 02 2005

Number of deco polyominoes of height n+1 having no reentrant corners along the lower contour (i.e. no vertical step that is followed by a horizontal step). In other words, a(n)=A121579(n+1,0). 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(1)=2 because because the only deco polyominoes of height 2 are the vertical and horizontal dominoes, having no reentrant corners along their lower contours. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 16 2006

Unreduced numerators of partial sums of the Taylor series for e. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 18 2006

a(n) = number of permutations on [n+1] (written in one-line notation) for which the subsequence beginning at 1 is increasing. Example: a(2)=5 counts 123, 213, 231, 312, 321. - David Callan (callan(AT)stat.wisc.edu), Oct 06 2006

a(n) and (1,-2,3,-4,5,-6,7,...) form a reciprocal pair under the list partition transform and associated operations described in A133314. - Tom Copeland (tcjpn(AT)msn.com), Nov 01 2007

Consider the subsets of the set {1,2,3,...,n} formed by the first n integers. E.g. for n = 3 we have {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. Let the variable sbst denote a subset. For each subset sbst we determine its number of parts, that is nprts(sbst). The sum over all possible subsets is written as sum_{sbst=subsets}. Then a(n) = sum_{sbst=subsets} nprts(sbst)!. E.g. for n = 3 we have 1!+1!+1!+1!+2!+2!+2!+3!=16. - Thomas Wieder (thomas.wieder(AT)t-online.de), Jun 17 2006

Equals row sums of triangle A158359(unsigned). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 17 2009]

Equals eigensequence of triangle A158821, Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 30 2009

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

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 75, Problem 9.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 65, p. 23, Ellipses, Paris 2008.

Bernd Gaertner, Walter D. Jr. Morris and Leo Ruest, Unique Sink Orientations of Grids, Algorithmica, Volume 51, Number 2 / June, 2008. [From N. J. A. Sloane, Jul 10 2009]

J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.

F. Luca and I. E. Shparlinski, On the square-free parts of [ e n! ], Glasgow Math. J., 49 (2007), 391-403.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 16.

D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.

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).

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.

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

P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, Boson normal ordering via substitutions and Sheffer-type polynomials

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 114

Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999) 138-150. (ps, pdf)

Lorenz Halbeisen and Saharon Shelah, Consequences of arithmetic for set theory, The Journal of Symbolic Logic, vol. 59 (1994), pp. 30-40.

M. Hassani, Counting and computing by e

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 35

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

T. Mansour and J. West, Avoiding 2-letter signed patterns.

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

J. Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463, ...: a surprising connection

J. Sondow, Which Partial Sums of the Taylor Series for e Are Convergents to e? (and a Link to the Primes $2, 5, 13, 37, 463, ...$) with an Appendix "Periodic Behaviour of Some Recurrence Sequences Related to $e$, Modulo Powers of 2" by Kyle Schalm

Eric Weisstein's World of Mathematics, Binomial Sums

Index entries for sequences related to logarithmic numbers

Index entries for related partition-counting sequences

Index entries for sequences related to factorial numbers

a(n) = n*a(n-1) + 1, a(0) = 1.

a(n) = n!*Sum(1/k!, k=0..n); a(n) = n!(e-Sum(1/k!, k=n+1...)).

a(0) = 1; for n > 0, a(n) = floor(e*n!).

E.g.f. exp(x)/(1-x).

a(n) = 1+sum{n >= k >= j >= 0}((k-j+1)*k!/j!) = a(n-1)+A001339(n-1) = A007526(n)+1. Binomial transformation of n!, i.e. A000142. - Henry Bottomley (se16(AT)btinternet.com), Jun 04 2001

Integral representation as n-th moment of a nonnegative function on a positive half-axis, in Maple notation: a(n)=exp(1)*int(x^n*exp(-x)*Heaviside(x-1), x=0..infinity), n=0, 1... - Karol A. Penson (penson(AT)lptl.jussieu.fr), Oct 01 2001

Formula, in Mathematica notation: Special values of Laguerre polynomials, a(n)=(-1)^n*n!*LaguerreL[n, -1-n, 1], n=1, 2... . This relation cannot be checked by Maple, as it appears that Maple does not incorporate Laguerre polynomials with second index equal to negative integers. It does check with Mathematica. - From Karol A. Penson ( penson(AT)lptl.jussieu.fr ) and Pawel Blasiak ( blasiak(AT)lptl.jussieu.fr), Feb 13 2004

G.f.: Sum_{k>=0} k!*x^k/(1-x)^(k+1). a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*k^(n-k)*(k+1)^k. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 18 2002

a(n) = Sum(k=0..n, A008290(n, k)*2^k ). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 12 2003

a(n) = Sum_{k = 0..n} A046716(n, k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 12 2004

Binomial transform of A000142; A000142(n) = n! . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 18 2004

a(n) = Sum[P(n, k), {k, 0, n}]. - Ross La Haye (rlahaye(AT)new.rr.com), Aug 28 2005

Given g.f. A(x), then g.f. A059115 = A(x/(1-x)). - Michael Somos Aug 03 2006

a(n) = 1+n+n(n-1)+...+n!. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 18 2006

a(n) = Integral_{0..infinity} (x+1)^n*exp(-x) dx - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 19 2006

Comments from Tom Copeland (tcjpn(AT)msn.com), Nov 01 2007, Dec 10 2007: (Start) Act on 1/(1-x) with 1/(1-xDx) = sum(j=0,1,...) (xDx)^j = sum(j=0,1,...) x^j*D^j*x^j = sum(j=0,1,...) j!*x^j*L(j,-:xD:,0) where Lag(n,x,0) are the Laguerre polynomials of order 0, D the derivative w.r.t. x and (:xD:)^j = x^j*D^j. Truncating the operator series at the j = n term gives an o.g.f. for a(0) through a(n) consistent with Jovovic's.

These results and those of Penson and Blasiak, Arnold, Bottomley and Deleham are related by the operator A094587 (the reverse of A008279), which is the umbral equivalent of n!*Lag[n,(.)!*Lag[.,x,-1],0] = (1-D)^(-1) x^n = (-1)^n * n! Lag(n,x,-1-n) = sum(j=0,...,n) Binom(n,j)*j!*x^(n-j) = sum(j=0,...,n) (n!/j!)*x^j . Umbral substitution of b(.) for x and then letting b(n)=1 for all n connects the results. See A132013 (the inverse of A094587) for a connection between these operations and 1/(1-xDx). (End)

Comments from Peter Bala (pbala(AT)toucansurf.com), Jul 15 2008 (Start): a(n)= n!*e - 1/(n + 1/(n+1 + 2/(n+2 + 3/(n+3 + ...)))).

Asymptotic result (Ramanujan): n!*e - a(n) ~ 1/n - 1/n^3 + 1/n^4 + 2/n^5 - 9/n^6 + ..., where the sequence [1,0,-1,1,2,-9,...] = [(-1)^k*A000587(k)], for k>=1.

a(n) is a difference divisibility sequence, that is, the difference a(n) - a(m) is divisible by n - m for all n and m (provided n is not equal to m). For fixed k, define the derived sequence a_k(n) := (a(n+k)-a(k))/n, n = 1,2,3,... . Then a_k(n) is also a difference divisibility sequence.

For example, the derived sequence a_0(n) is just a(n-1). The set of integer sequences satisfying the difference divisibility property forms a ring with term-wise operations of addition and multiplication.

Recurrence relations: a(0) = 1, a(n) = (n-1)*(a(n-1) + a(n-2)) + 2, for n >= 1. a(0) = 1, a(1) = 2, a(n) = (n+1)*a(n-1) - (n-1)*a(n-2) for n >= 2. The sequence b(n) := n! satisfies the latter recurrence with the initial conditions b(0) = 1, b(1) = 1. This leads to the finite continued fraction expansion a(n)/n! = 1/(1-1/(2-1/(3-2/(4-...-(n-1)/(n+1))))), n >= 2.

Lim n -> infinity a(n)/n! = e = 1/(1-1/(2-1/(3-2/(4-...-n/((n+2)-...))))). This is the particular case m = 0 of the general result m!/e - d_m = (-1)^(m+1) *(1/(m+2 -1/(m+3 -2/(m+4 -3/(m+5 -...))))), where d_m denotes the m_th derangement number A000166(m).

For sequences satisfying the more general recurrence a(n) = (n+1+r)*a(n-1) - (n-1)*a(n-2), which yield series acceleration formulas for e/r! that involve the Poisson-Charlier polynomials c_r(-n;-1), refer to A001339 (r=1), A082030 (r=2), A095000 (r=3) and A095177 (r=4).

For the corresponding results for the constants log(2), zeta(2) and zeta(3) refer to A142992, A108625 and A143007 respectively. (End)

G.f. satisfies: A(x) = 1/(1-x)^2 + x^2*A'(x)/(1-x). [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 03 2008]

With two objects we can form 5 sequences: (), (a), (b), (a,b), (b,a), so a(2) = 5.

A000522 := n->add(n!/k!, k=0..n);

restart: G(x):=exp(x)/(1-x): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..20); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 03 2009]

Table[ Gamma[ n, 1 ]*E, {n, 1, 24} ]; FunctionExpand[ % ]

...and/or... s=2; lst={1, s}; Do[s+=s++n; AppendTo[lst, s], {n, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 23 2008]

(PARI) a(n)=local(A); if(n<0, 0, A=vector(n+1); A[1]=1; for(k=1, n, A[k+1]=k*A[k]+1); A[n+1]) /* Michael Somos Jul 1 2004 */

(PARI) a(n)=if(n<0, 0, n!*polcoeff(exp(x+x*O(x^n))/(1-x), n)) - Michael Somos Mar 06 2004

(PARI) a(n)=floor(exp(1)*(n)! - 1/(n +1 )) - Alexander R.Povolotsky (pevnev(AT)juno.com), Nov 05 2007

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x)^2+x^2*deriv(A)/(1-x)); polcoeff(A, n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 03 2008]

Cf. A010844, A010845, A038159, A002627, A006231, A000166, A072453, A072456, A008290.

Cf. A007526, A054091, A073591, A001339, A082030, A095000, A095177, A142992, A108625, A143007, A064383, A064384, A124779, A121579.

Average of n-th row of triangle in A007526.

Row sums of A008279.

First differences give A001339. Second differences give A001340.

a(n) = A007526(n-1) + 1.

a(n)=[2/(n+1)]A009578(n+1)-1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 24 2001

Partial sums are in A001338, A002104.

a(n) = A061354(n)*A093101(n).

a(n)=sum(A094816(n, k), k=0..n), n>=0 (row sums of Poisson-Charlier coefficient matrix).

A row of the array in A144502.

Cf. A158359, A158821. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 17 2009]

Sequence in context: A131178 A003149 A027046 this_sequence A007469 A091139 A084785

Adjacent sequences: A000519 A000520 A000521 this_sequence A000523 A000524 A000525

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Michael Somos.

There are some related sequences mentioned by Luca and Shparlinski that it would be nice to have in the OEIS, if someone would add them. For n >= 1, write a(n) = s(n) u(n)^2, where s(n) is squarefree. Then it would be nice to have the sequences s(n), S(n) = Product_{i <= n} s(i), u(n) and possibly t(n) = number of i <= n such that floor( e i! ) is a square. - N. J. A. Sloane (njas(AT)research.att.com), Oct 29 2007