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A001286 Lah numbers: (n-1)*n!/2.
(Formerly M4225 N1766)
50
1, 6, 36, 240, 1800, 15120, 141120, 1451520, 16329600, 199584000, 2634508800, 37362124800, 566658892800, 9153720576000, 156920924160000, 2845499424768000, 54420176498688000, 1094805903679488000, 23112569077678080000 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

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, 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, Apr 03 2004

First Eulerian transform of 0,1,2,3,4,... . - Ross La Haye, Mar 05 2005

With offset 0 : determinant of the n X n matrix m(i,j)=(i+j+1)!/i!/j!. - Benoit Cloitre, 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, 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. - Emeric Deutsch, Dec 15 2008

(-1)^(n+1)*a(n) is the determinant of the n X n matrix whose (i,j)-th element is 0 for i = j, is j-1 for j>i, and j for j < i. - Michel Lagneau, May 04 2010

Row sums of the triangle in A030298. - Reinhard Zumkeller, Mar 29 2012

a(n) is the total number of ascents (descents) over all n-permutations. a(n) = Sum_{k=1,...,n} A008292(n,k)*k. - Geoffrey Critzer, Jan 06 2013

For m>=4, a(m-2) is the number of Hamiltonian cycles in a simple graph with m vertices which is complete, except for one edge. Proof: think of distinct round-table seatings of m persons such that persons "1" and "2" may not be neighbors; the count is (m-3)(m-2)!/2. See also A001710. - Stanislav Sykora, Jun 17 2014

REFERENCES

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.

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.

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).

LINKS

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 399

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Sandi Klavžar, Uroš Milutinović and Ciril Petr, 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.

Eric Weisstein's World of Mathematics, Permutation Ascent

FORMULA

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

Row sums of table A051683. - Alford Arnold, Sep 29 2006

5-th binomial transform of A135218: (1, 1, 1, 25, 25, 745, 3145,...). - Gary W. Adamson, 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). - Milan Janjic, Mar 01 2009

a(n) = A000217(n-1)*A000142(n-1). - Reinhard Zumkeller, May 15 2010

a(n) = (n+1)!*sum(1/(k^2+3*k+2),k=1..n-1). - _Gary Detlefs, Sep 14 2011

EXAMPLE

a(10) = (1+2+3+4+5+6+7+8+9)*(1*2*3*4*5*6*7*8*9) = 16329600. - Reinhard Zumkeller, May 15 2010

MAPLE

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

seq(sum(mul(j, j=3..n), k=2..n), n=2..21); # Zerinvary Lajos, Jun 01 2007

a:=n->sum(k*mul(k, k=1..n), k=1..n):seq(a(n), n=1...19); # Zerinvary Lajos, 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); # Zerinvary Lajos, Apr 01 2009]

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

MATHEMATICA

lst={}; s=0; Do[s=s+n; AppendTo[lst, n!*s], {n, 30}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 07 2008 *)

Table[Sum[n!, {i, 2, n}]/2, {n, 2, 20}] (* Zerinvary Lajos, Jul 12 2009 *)

nn=20; With[{a=Accumulate[Range[nn]], t=Range[nn]!}, Times@@@Thread[{a, t}]] (* Harvey P. Dale, Jan 26 2013 *)

PROG

(Sage) [(n-1)*factorial(n)/2 for n in xrange(2, 21)] # Zerinvary Lajos, May 16 2009

(Haskell)

a001286 n = sum[1..n-1] * product [1..n-1]

-- _Reinhard Zumkeler_, Aug 01 2011

(Maxima) A001286(n):=(n-1)*n!/2$

makelist(A001286(n), n, 1, 30); /* Martin Ettl, Nov 03 2012 */

(PARI) a(n)=(n-1)*n!/2 \\ Charles R Greathouse IV, Nov 20 2012

CROSSREFS

Cf. A001710, A052609, A062119, A075181, A060638, A060608, A060570, A060612, A135218, A019538, A053495, A051683.

A002868 is an essentially identical sequence.

Column 2 of |A008297|.

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

Sequence in context: A066053 A153824 A180119 * A181964 A199422 A049431

Adjacent sequences:  A001283 A001284 A001285 * A001287 A001288 A001289

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 24 16:46 EST 2014. Contains 249899 sequences.