0,3

A similar sequence, with the initial 0 replaced by 1, namely A094258, is defined by the recurrence a(2) = 1, a(n) = a(n-1)*(n-1)^2/(n-2). - Andrey Ryshevich (ryshevich(AT)notes.idlab.net), May 21 2002

Denominators in power series expansion of E_1(x)+gamma+log(x), n>0.

If all the permutations of any length k are arranged in lexicographic order, the n-th term in this sequence (n <= k) gives the index of the permutation that rotates the last n elements one position to the right. E.g. there are 24 permutations of 4 items. In lexicographic order they are (0,1,2,3), (0,1,3,2), (0,2,1,3),... (3,2,0,1), (3,2,1,0). Permutation 0 is (0,1,2,3) which rotates the last 1 element, i.e. is makes no change. Permutation 1 is (0,1,3,2) which rotates the last 2 element. Pwermutation 4 is (0,3,1,2) which rotates the last 3 elements. Permutation 18 is (3,0,1,2) which rotates the last 4 elements. The same numbers work for permutations of any length. - Henry H. Rich (glasss(AT)bellsouth.net), Sep 27 2003

Stirling transform of a(n+1)=[4,18,96,600,...] is A083140(n+1)=[4,22,154,...]. - Michael Somos Mar 04 2004

Number of small descents in all permutations of [n+1]. A small descent in a permutation (x_1,x_2,...,x_n) is a position i such that x_i - x_(i+1) =1. Example: a(2)=4 because there are 4 small descents in the permutations 123, 13\2, 2\13, 231, 312, 3\2\1 of {1,2,3} (shown by \). a(n)=Sum(k*A123513(n,k), k=0..n-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 02 2006

a(n-1) is the number of permutations of n in which n is not fixed; equivalently, the number of permutations of the positive integers in which n is the largest element that is not fixed. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 29 2006

Number of factors in a determinant when writing down all multiplication permutations. [From Mats O. Granvik (mgranvik(AT)abo.fi), Sep 12 2008]

a(n) is also the sum of the positions of the left-to-right maxima in all permutations of [n]. Example: a(3)=18 because the positions of the left-to-right maxima in the permutations 123,132,213,231,312 and 321 of [3] are 123, 12, 13, 12, 1 and 1, respectively and 1+2+3+1+2+1+3+1+2+1+1=18. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 21 2008]

Equals eigensequence of triangle A002024 ("n appears n times". [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 29 2008]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 17 2009: (Start)

Preface the series with another 1: (1, 1, 4, 18,...); then the next term =

dot product of the latter with "n occurs n times". Example: 96 = (1, 1, 4, 8)

dot (4, 4, 4, 4) = (4 + 4 + 16 + 72). (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 218.

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 336.

A. van Heemert, Cyclic permutations with sequences and related problems, J. Reine Angew. Math., 198 (1957), 56-72.

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

I. Kortchemski, Asymptotic behavior of permutation records, arXiv: 0804.0446v2 [math.CO], 18 May 2008. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 21 2008]

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

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 30

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

E.g.f.: x/(1-x)^2. a(n)=-A021009(n, 1), n >= 0.

The coefficient of y^(n-1) in expansion of (y+n!)^n, n >= 1, gives the sequence 1, 4, 18, 96, 600, 4320, 35280, ... - Artur Jasinski (grafix(AT)csl.pl), Oct 22 2007

Integral representation as n-th moment of a function on a positive half-axis, in Maple notation: a(n)=int(x^n*(x*(x-1)*exp(-x)), x=0..infinity), n=0, 1... This representation may not be unique. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Sep 27 2001

a(0)=0, a(n)=n*a(n-1)+n! - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 16 2003

a(0) = 0, a(n) = (n - 1) * (1 + Sum i=1..n-1 a(i)) for i > 0 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 11 2004

Arises in the denominators of the following identities: Sum_{1..oo}1/(n(n+1)(n+2)) = 1/4, Sum_{1..oo}1/(n(n+1)(n+2)(n+3)) = 1/18, Sum_{1..oo}1/(n(n+1)(n+2)(n+3)(n+4)) = 1/96, etc. The general expression is Sum_{n = k..infinity } 1/C(n, k) = k/(k-1). - Dick Boland, Jun 06 2005

a(n)= sum(|Stirling1(n+1, m)|, m=2..n+1), n>=1 and a(0):=0, where Stirling1(n, m)= A048994(n, m), n>=>m=0.

a(n) = 1/sum(k!/(n+k+1)!,k=0..infinity), n>0. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 13 2006

a(n)=Sum(k*A143946(n,k),k=1..n(n+1)/2). [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 21 2008]

The reciprocals of a(n) are the lead coefficients in the factored form of the polynomials obtained by summing the binomial coefficients with a fixed lower term up to n as the upper term, divided by the term index, for n >= 1: sum_{k = i..n} C(k, i)/k = (1/a(n))*n*(n-1)*..*(n-i+1). The first few such polynomials are: sum_{k = 1..n} C(k, 1)/k = (1/1)*n, sum_{k = 2..n} C(k, 2)/k = (1/4)*n*(n-1), sum_{k = 3..n} C(k, 3)/k = (1/18)*n*(n-1)*(n-2), sum_{k = 4..n} C(k, 4)/k = (1/96)*n*(n-1)*(n-2)*(n-3) etc. [From Peter Breznay (breznayp(AT)uwgb.edu), Sep 28 2008]

If we define f(n,i,x)= sum(sum(binomial(k,j)*stirling1(n,k)*stirling2(j,i)*x^(k-j),j=i..k),k=i..n) then a(n)=(-1)^(n-1)*f(n,1,-2), (n>=1). [From Milan R. Janjic (agnus(AT)blic.net), Mar 01 2009]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

a(0) = 0, a(1) = 1 and a(n) = (1+(n-1))^2*product((1+k), k=1..n-2), n=>2.

(End)

E_1(x)+gamma+log(x)=x/1-x^2/4+x^3/18-x^4/96+..., x>0.

A001563 := n->n*n!;

seq(sum(mul(j, j=1..n), k=1..n), n=0..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 01 2007

a:=n->sum(sum(stirling1(n, k), j=2..n), k=2..n): seq(abs(a(n)), n=1..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 28 2007

spec := [S, {S = Union(Prod(Union(Z, Z, Z), Sequence(Z), Sequence(Z)), Prod(Union(Z, Z), Sequence(Z), Sequence(Z)))}, labelled]: seq(combstruct[count](spec, size=n)/5, n=0..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2008

with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(1):seq(count(ZLL, size=n)*n, n=0..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 11 2008

a:=n->add((n!), j=1..n):seq(a(n), n=0..21); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 27 2008]

restart: G(x):=x/(1-x)^2: f[0]:=G(x): for n from 1 to 24 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..21); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 01 2009]

restart:printlevel := -1; a := [0]; T := x->LambertW(-x); f := series((T(x)*(1+T(x)))/(1-T(x)), x, 24); for m from 1 to 21 do a := [op(a), op(2*m, f)*m! ] od; print(a); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2009]

a = {}; Do[k = CoefficientList[Expand[(y + n!)^n], y]; AppendTo[a, k[[Length[k] - 1]]], {n, 1, 50}]; a - Artur Jasinski (grafix(AT)csl.pl), Oct 22 2007

..and/or..a=s=1; lst={}; Do[a=s*n-s; s+=a; AppendTo[lst, a], {n, 2*4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 27 2009]

Table[Sum[n!, {i, 1, n}], {n, 0, 21}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]

Table[(-1)^(n + 1)* Sum[(-1)^(n - k) k (-1)^(n - k) StirlingS1[n + 2, k + 2], {k, 0, n}], {n, 0, 21}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009]

(PARI) a(n)=if(n<0, 0, n*n!)

Cf. A047920, A047922, A000142, A055089, A053495.

Cf. A123513.

A143946 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 21 2008]

A002024 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 29 2008]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

Cf. A163931 (E(x,m,n)), A002775 (n^2*n!), A091363 (n^3*n!), A091364 (n^4*n!).

(End)

Sequence in context: A081103 A005777 A152392 this_sequence A094304 A094258 A086681

Adjacent sequences: A001560 A001561 A001562 this_sequence A001564 A001565 A001566

nonn,easy,nice

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