Search: id:A000522 Results 1-1 of 1 results found. %I A000522 M1497 N0589 %S A000522 1,2,5,16,65,326,1957,13700,109601,986410,9864101,108505112,1302061345, %T A000522 16926797486,236975164805,3554627472076,56874039553217,966858672404690, %U A000522 17403456103284421,330665665962404000,6613313319248080001 %N A000522 Total number of arrangements of a set with n elements: a(n) = Sum_{k=0..n} n!/k!. %C A000522 The number of one-to-one sequences that can be formed from n distinct objects. %C A000522 Related to number of operations of addition and multiplication to evaluate a determinant of order n by cofactor expansion - see A026243. %C A000522 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 %C A000522 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 %C A000522 a(n) is the permanent of the n X n matrix with 2s on the diagonal and 1s elsewhere. - Yuval Dekel, Nov 01 2003 %C A000522 (A000166 + this_sequence)/2 = A009179, (A000166 - this_sequence)/2 = A009628. %C A000522 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 %C A000522 Number of {12,12*,21*}- and {12,12*,2*1}-avoiding signed permutations in the hyperoctahedral group. %C A000522 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 %C A000522 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 %C A000522 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 %C A000522 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 %C A000522 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 %C A000522 Unreduced numerators of partial sums of the Taylor series for e. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 18 2006 %C A000522 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 %C A000522 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 %C A000522 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 %C A000522 Equals row sums of triangle A158359(unsigned). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 17 2009] %C A000522 Equals eigensequence of triangle A158821, Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 30 2009 %D A000522 E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42. %D A000522 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 75, Problem 9. %D A000522 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 65, p. 23, Ellipses, Paris 2008. %D A000522 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] %D A000522 J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. %D A000522 F. Luca and I. E. Shparlinski, On the square-free parts of [ e n! ], Glasgow Math. J., 49 (2007), 391-403. %D A000522 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 16. %D A000522 D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70. %D A000522 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000522 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A000522 J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641. %H A000522 T. D. Noe, Table of n, a(n) for n=0..100 %H A000522 P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, Boson normal ordering via substitutions and Sheffer-type polynomials %H A000522 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A000522 P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 114 %H A000522 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) %H A000522 Lorenz Halbeisen and Saharon Shelah, Consequences of arithmetic for set theory, The Journal of Symbolic Logic, vol. 59 (1994), pp. 30-40. %H A000522 M. Hassani, Counting and computing by e %H A000522 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 35 %H A000522 J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5. %H A000522 T. Mansour and J. West, Avoiding 2-letter signed patterns. %H A000522 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 %H A000522 J. Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463, ...: a surprising connection %H A000522 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 %H A000522 Eric Weisstein's World of Mathematics, Binomial Sums %H A000522 Index entries for sequences related to logarithmic numbers %H A000522 Index entries for related partition-counting sequences %H A000522 Index entries for sequences related to factorial numbers %F A000522 a(n) = n*a(n-1) + 1, a(0) = 1. %F A000522 a(n) = n!*Sum(1/k!, k=0..n); a(n) = n!(e-Sum(1/k!, k=n+1...)). %F A000522 a(0) = 1; for n > 0, a(n) = floor(e*n!). %F A000522 E.g.f. exp(x)/(1-x). %F A000522 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 %F A000522 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 %F A000522 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 %F A000522 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 %F A000522 a(n) = Sum(k=0..n, A008290(n, k)*2^k ). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 12 2003 %F A000522 a(n) = Sum_{k = 0..n} A046716(n, k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 12 2004 %F A000522 Binomial transform of A000142; A000142(n) = n! . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 18 2004 %F A000522 a(n) = Sum[P(n, k), {k, 0, n}]. - Ross La Haye (rlahaye(AT)new.rr.com), Aug 28 2005 %F A000522 Given g.f. A(x), then g.f. A059115 = A(x/(1-x)). - Michael Somos Aug 03 2006 %F A000522 a(n) = 1+n+n(n-1)+...+n!. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 18 2006 %F A000522 a(n) = Integral_{0..infinity} (x+1)^n*exp(-x) dx - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 19 2006 %F A000522 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. %F A000522 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) %F A000522 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 + ...)))). %F A000522 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. %F A000522 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. %F A000522 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. %F A000522 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. %F A000522 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). %F A000522 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). %F A000522 For the corresponding results for the constants log(2), zeta(2) and zeta(3) refer to A142992, A108625 and A143007 respectively. (End) %F A000522 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] %e A000522 With two objects we can form 5 sequences: (), (a), (b), (a,b), (b,a), so a(2) = 5. %p A000522 A000522 := n->add(n!/k!,k=0..n); %p A000522 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] %t A000522 Table[ Gamma[ n, 1 ]*E, {n, 1, 24} ]; FunctionExpand[ % ] %t A000522 ...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] %o A000522 (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 */ %o A000522 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(x+x*O(x^n))/(1-x),n)) - Michael Somos Mar 06 2004 %o A000522 (PARI) a(n)=floor(exp(1)*(n)! - 1/(n +1 )) - Alexander R.Povolotsky (pevnev(AT)juno.com), Nov 05 2007 %o A000522 (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] %Y A000522 Cf. A010844, A010845, A038159, A002627, A006231, A000166, A072453, A072456, A008290. %Y A000522 Cf. A007526, A054091, A073591, A001339, A082030, A095000, A095177, A142992, A108625, A143007, A064383, A064384, A124779, A121579. %Y A000522 Average of n-th row of triangle in A007526. %Y A000522 Row sums of A008279. %Y A000522 First differences give A001339. Second differences give A001340. %Y A000522 a(n) = A007526(n-1) + 1. %Y A000522 a(n)=[2/(n+1)]A009578(n+1)-1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 24 2001 %Y A000522 Partial sums are in A001338, A002104. %Y A000522 a(n) = A061354(n)*A093101(n). %Y A000522 a(n)=sum(A094816(n, k), k=0..n), n>=0 (row sums of Poisson-Charlier coefficient matrix). %Y A000522 A row of the array in A144502. %Y A000522 Cf. A158359, A158821. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 17 2009] %Y A000522 Sequence in context: A131178 A003149 A027046 this_sequence A007469 A091139 A084785 %Y A000522 Adjacent sequences: A000519 A000520 A000521 this_sequence A000523 A000524 A000525 %K A000522 nonn,easy,nice %O A000522 0,2 %A A000522 N. J. A. Sloane (njas(AT)research.att.com). %E A000522 Additional comments from Michael Somos. %E A000522 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 Search completed in 0.003 seconds