2,2

Sum((-1)^i * i^(n+1) * binomial( n, i), i=0..n) = (-1)^n * n * (n+1)! /2 - Yong Kong (ykong(AT)curagen.com), Dec 26 2000

Number of surjections from {1,...,n} to {1,....,n-1} - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 05 2003

a(n+1)=(-1)^(n+1)*det(M_n) where M_n is the n X n matrix M_(i,j)=max(i*(i+1)/2,j*(j+1)/2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 03 2004

First Eulerian transform of 0,1,2,3,4... - Ross La Haye (rlahaye(AT)new.rr.com), Mar 05 2005

With offset 0 : determinant of the n X n matrix m(i,j)=(i+j+1)!/i!/j! - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 11 2005

Comment from Alexander Povolotsky, Oct 16 2006. These numbers arise when expressing n(n+1)(n+2)..(n+k)[n+(n+1)+(n+2)+..+(n+k)] as sums of squares: n(n+1)[n+(n+1)] = 6(1+4+9+16+ .. + n^2), n(n+1)(n+2)[n+(n+1)+(n+2)]=36(1+(1+4)+(1+4+9)+..+(1+4+9+16+ .. + n^2)), n(n+1)(n+2)(n+3)[n+(n+1)+(n+2)+(n+3)]= 240(....), ...

a(n) = number of edges in the Hasse diagram for the weak Bruhat order on the symmetric group S_n. For permutations p,q in S_n, q covers p in the weak Bruhat order if p,q differ by an adjacent transposition and q has one more inversion than p. Thus 23514 covers 23154 due to the transposition that interchanges the third and fourth entries. Cf. A002538 for the strong Bruhat order. - David Callan (callan(AT)stat.wisc.edu), Nov 29 2007

a(n) is also the number of excedances in all permutations of {1,2,...,n} (an excedance of a permutation p is a value j such p(j)>j). Proof: j is exceeded (n-1)! times by each of the numbers j+1, j+2, ..., n; now, Sum[(n-j)(n-1)!,j=1..n)=n!(n-1)/2. Example: a(3)=6 because the number of excedances of the permutations 123, 132, 312, 213, 231, 321 are 0, 1, 1, 1, 2, 1, respectively. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 2008]

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, p. 90, ex. 4.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.

Klavzar, S.; Milutinovic, U.; and Petr, C., Hanoi graphs and some classical numbers, Expo. Math. 23 (2005), no. 4, 371-378.

S. Lehr, J. Shallit and J. Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.

A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.

T. D. Noe, Table of n, a(n) for n=2..100

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 399

Eric Weisstein's World of Mathematics, Permutation Ascent

E.g.f.: x^2/[2(1-x)^2]. - Ralf Stephan, Apr 02 2004

Row sums of table A051683. - Alford Arnold (Alford1940(AT)aol.com), Sep 29 2006

5-th binomial transform of A135218: (1, 1, 1, 25, 25, 745, 3145,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 23 2007

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*f(n,2,-2), (n>=2). [From Milan R. Janjic (agnus(AT)blic.net), Mar 01 2009]

a:=n->sum(j*n!, j=0..n): seq(a(n), n=0..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 01 2006

seq(sum(mul(j, j=3..n), k=2..n), n=2..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 01 2007

a:=n->sum(k*mul(k, k=1..n), k=1..n):seq(a(n), n=1...19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 11 2008

restart: G(x):=x^2/(1-x)^2: f[0]:=G(x): for n from 1 to 20 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/2, n=2..20); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 01 2009]

with(combinat):seq(n/2*numbperm(n+1, n), n=1..19); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 17 2009]

lst={}; s=0; Do[s=s+n; AppendTo[lst, n!*s], {n, 30}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 07 2008]

Table[Sum[n!, {i, 2, n}]/2, {n, 2, 20}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]

(Other) sage: [(n-1)*factorial(n)/2 for n in xrange(2, 21)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]

Cf. A052609, A062119, A075181, A060638, A060608, A060570, A060612.

A002868 is an essentially identical sequence.

Column 2 of |A008297|. Cf. A019538, A053495, A051683.

Third column (m=2) of triangle |A111596(n, m)|: matrix product of |S1|.S2 Stirling number matrices.

Cf. A135218.

Sequence in context: A057395 A066053 A153824 this_sequence A049431 A049428 A129063

Adjacent sequences: A001283 A001284 A001285 this_sequence A001287 A001288 A001289

nonn,easy,nice

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