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A001286
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Lah numbers: (n-1)*n!/2.
(Formerly M4225 N1766)
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49
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1, 6, 36, 240, 1800, 15120, 141120, 1451520, 16329600, 199584000, 2634508800, 37362124800, 566658892800, 9153720576000, 156920924160000, 2845499424768000, 54420176498688000, 1094805903679488000, 23112569077678080000
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OFFSET
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2,2
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COMMENTS
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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. [From 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
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REFERENCES
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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.
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).
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LINKS
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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
Eric Weisstein's World of Mathematics, Permutation Ascent
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FORMULA
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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). [From Milan Janjic, Mar 01 2009]
a(n) = A000217(n-1)*A000142(n-1). [From Reinhard Zumkeller, May 15 2010]
a(n)= (n+1)!*sum(1/(k^2+3*k+2),k=1..n-1). [From Gary Detlefs, Sep 14 2011]
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EXAMPLE
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a(10) = (1+2+3+4+5+6+7+8+9)*(1*2*3*4*5*6*7*8*9) = 16329600. [From Reinhard Zumkeller, May 15 2010]
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MAPLE
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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); # [From Zerinvary Lajos, Apr 01 2009]
with(combinat):seq(n/2*numbperm(n+1, n), n=1..19); # [From Zerinvary Lajos, Apr 17 2009]
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MATHEMATICA
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lst={}; s=0; Do[s=s+n; AppendTo[lst, n!*s], {n, 30}]; lst [From Vladimir Joseph Stephan Orlovsky, Sep 07 2008]
Table[Sum[n!, {i, 2, n}]/2, {n, 2, 20}] [From 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 *)
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PROG
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(Sage) [(n-1)*factorial(n)/2 for n in xrange(2, 21)] # [From 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
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CROSSREFS
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Cf. 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
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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