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

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.

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.yu), Sep 13 2006

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

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

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

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

Cf. A123513.

Adjacent sequences: A001560 A001561 A001562 this_sequence A001564 A001565 A001566

Sequence in context: A138901 A109074 A134357 this_sequence A094304 A094258 A086681

nonn,easy,nice

njas