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 A005990 a(n) = (n-1)*(n+1)!/6. (Formerly M4551) 19
 0, 1, 8, 60, 480, 4200, 40320, 423360, 4838400, 59875200, 798336000, 11416204800, 174356582400, 2833294464000, 48819843072000, 889218570240000, 17072996548608000, 344661117825024000, 7298706024529920000, 161787983543746560000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Coefficients of Gandhi polynomials. a(n) = Sum_{pi in Symm(n)} Sum_{i=1..n} max(pi(i)-i,0), i.e., the total positive displacement of all letters in all permutations on n letters. - Franklin T. Adams-Watters, Oct 25 2006 a(n) is also the sum of the excedances of all permutations of [n]. An excedance of a permutation p of [n] is an i (1 <= i <= n-1) such that p(i) > i. Proof: i is an excedance if p(i) = i+1, i+2, ..., n (n-i possibilities), with the remaining values of p forming any permutation of [n]\{p(i)} in the positions [n]\{i} ((n-1)! possibilities). Summation of i(n-i)(n-1)! over i from 1 to n-1 completes the proof. Example: a(3)=8 because the permutations 123, 132, 213, 231, 312, 321 have excedances NONE, {2}, {1}, {1,2}, {1}, {1}, respectively. - Emeric Deutsch, Oct 26 2008 a(n) is also the number of doubledescents in all permutations of {1,2,...,n-1}. We say that i is a doubledescent of a permutation p if p(i) > p(i+1) > p(i+2). Example: a(3)=8 because each of the permutations 1432, 4312, 4213, 2431, 3214, 3421 has one doubledescent, the permutation 4321 has two doubledescents and the remaining 17 permutations of {1,2,3,4} have no doubledescents. - Emeric Deutsch, Jul 26 2009 Half of sum of abs(p(i+1) - p(i)) over all permutations on n, e.g., 42531 = 2 + 3 + 2 + 2 = 9, and the total over all permutations on {1,2,3,4,5} is 960. - Jon Perry, May 24 2013 a(n) gives the number of non-occupied corners in tree-like tableaux of size n+1 (see Gao et al. link). - Michel Marcus, Nov 18 2015 a(n) is the number of sequences of n+2 balls colored with at most n colors such that exactly three balls are the same color as some other ball in the sequence. - Jeremy Dover, Sep 26 2017 a(n) is the number of triangles (3-cycles) in the (n+1)-alternating group graph. - Eric W. Weisstein, Jun 09 2019 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..300 D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy) Alice L. L. Gao, Emily X. L. Gao, Brian Y. Sun, Zubieta's Conjecture on the Enumeration of Corners in Tree-like Tableaux, arXiv:1511.05434 [math.CO], 2015. The second version of this paper has a different title and different authors: A. L. L. Gao, E. X. L. Gao, P. Laborde-Zubieta, and B. Y. Sun, Enumeration of Corners in Tree-like Tableaux and a Conjectural (a,b)-analogue, arXiv preprint arXiv:1511.05434v2, 2015. Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets Eric Weisstein's World of Mathematics, Alternating Group Graph Eric Weisstein's World of Mathematics, Graph Cycle FORMULA a(n) = A052571(n+2)/6. - Zerinvary Lajos, May 11 2007 a(n) = Sum_{m=0..n} Sum_{k=-1..n} Sum_{j=1..n} n!/6, n >= 0. - Zerinvary Lajos, May 11 2007 If we define f(n,i,x) = Sum_{k=i..n} (Sum_{j=i..k} binomial(k,j)*stirling1(n,k)*stirling2(j,i)*x^(k-j)) then a(n+1) = (-1)^(n-1)*f(n,1,-4), (n >= 1). - Milan Janjic, Mar 01 2009 E.g.f.: (-1+3*x)/(3!*(1-x)^3), a(0) = -1/3!. Such e.g.f. computations resulted from e-mail exchange with Gary Detlefs. - Wolfdieter Lang, May 27 2010 a(n) = ((n+3)!/2) * Sum_{j=i..k} (k+1)!/(k+3)!, with offset 0. - Gary Detlefs, Aug 05 2010 a(n) = (n+2)!*Sum_{k=1..n-1} 1/((2*k+4)*(k+3)). - Gary Detlefs, Oct 09 2011 a(n) = (n+2)!*(1 + 3*(H(n+1) - H(n+2)))/6, where H(n) is the n-th harmonic number. - Gary Detlefs, Oct 09 2011 With offset = 0, e.g.f.: x/(1-x)^4. - Geoffrey Critzer, Aug 30 2013 MAPLE [ seq((n-1)*(n+1)!/6, n=1..40) ]; a:=n->sum(sum(sum(n!/6, j=1..n), k=-1..n), m=0..n): seq(a(n), n=0..19); # Zerinvary Lajos, May 11 2007 seq(sum(mul(j, j=3..n), k=3..n)/3, n=2..21); # Zerinvary Lajos, Jun 01 2007 restart: G(x):=x^3/(1-x)^2: f:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/3!, n=2..21); # Zerinvary Lajos, Apr 01 2009 MATHEMATICA Table[Sum[n!/6, {i, 3, n}], {n, 2, 21}] (* Zerinvary Lajos, Jul 12 2009 *) Table[(n - 1) (n + 1)!/6, {n, 20}] (* Harvey P. Dale, Apr 07 2019 *) Table[(n - 1) Pochhammer[4, n - 2], {n, 20}] (* Eric W. Weisstein, Jun 09 2019 *) Table[(n - 1) Gamma[n + 2]/6, {n, 20}] (* Eric W. Weisstein, Jun 09 2019 *) Range[0, 20]! CoefficientList[Series[x/(1 - x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Jun 09 2019 *) PROG (MAGMA) [(n-1)*Factorial(n+1)/6: n in [1..25]]; // Vincenzo Librandi, Oct 11 2011 (PARI) a(n)=(n-1)*(n+1)!/6 \\ Charles R Greathouse IV, May 24 2013 CROSSREFS A090672(n)/2. Cf. A001715. Equals the second right hand column of A167568 divided by 2. - Johannes W. Meijer, Nov 12 2009 Sequence in context: A233666 A199526 A129331 * A160228 A274746 A317774 Adjacent sequences:  A005987 A005988 A005989 * A005991 A005992 A005993 KEYWORD nonn,easy AUTHOR EXTENSIONS Better definition from Robert Newstedt STATUS approved

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Last modified February 17 17:59 EST 2020. Contains 331999 sequences. (Running on oeis4.)