

A005990


a(n) = (n1)*(n+1)!/6.
(Formerly M4551)


18



0, 1, 8, 60, 480, 4200, 40320, 423360, 4838400, 59875200, 798336000, 11416204800, 174356582400, 2833294464000, 48819843072000, 889218570240000, 17072996548608000, 344661117825024000, 7298706024529920000, 161787983543746560000
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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. AdamsWatters, 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 <= n1) such that p(i) > i. Proof: i is an excedance if p(i) = i+1, i+2, ..., n (ni possibilities), with the remaining values of p forming any permutation of [n]\{p(i)} in the positions [n]\{i} ((n1)! possibilities). Summation of i(ni)(n1)! over i from 1 to n1 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,...,n1}. 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 nonoccupied corners in treelike 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


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), 305318.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305318. (Annotated scanned copy)
Alice L. L. Gao, Emily X. L. Gao, Brian Y. Sun, Zubieta's Conjecture on the Enumeration of Corners in Treelike 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. LabordeZubieta, and B. Y. Sun, Enumeration of Corners in Treelike Tableaux and a Conjectural (a,b)analogue, arXiv preprint arXiv:1511.05434v2, 2015.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets


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^(kj)) then a(n+1) = (1)^(n1)*f(n,1,4), (n >= 1).  Milan Janjic, Mar 01 2009
E.g.f.: (1+3*x)/(3!*(1x)^3), a(0) = 1/3!. Such e.g.f. computations resulted from email 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..n1} 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 nth harmonic number.  Gary Detlefs, Oct 09 2011
With offset = 0, e.g.f.: x/(1x)^4.  Geoffrey Critzer, Aug 30 2013


MAPLE

[ seq((n1)*(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/(1x)^2: f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n1], 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 *)


PROG

(MAGMA) [(n1)*Factorial(n+1)/6: n in [1..25]]; // Vincenzo Librandi, Oct 11 2011
(PARI) a(n)=(n1)*(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

N. J. A. Sloane


EXTENSIONS

Better definition from Robert Newstedt


STATUS

approved



