

A090672


a(n) = (n^21)*n!/3.


7



0, 2, 16, 120, 960, 8400, 80640, 846720, 9676800, 119750400, 1596672000, 22832409600, 348713164800, 5666588928000, 97639686144000, 1778437140480000, 34145993097216000, 689322235650048000, 14597412049059840000, 323575967087493120000, 7493338185184051200000
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OFFSET

1,2


COMMENTS

a(n) = Sum_{pi in Symm(n)} Sum_{i=1..n} pi(i)i, i.e., the total displacement of all letters in all permutations on n letters.
a(n) = number of entries between the entries 1 and 2 in all permutations of {1,2,...,n+1}. Example: a(2)=2 because we have 123, 1(3)2, 213, 2(3)1, 312, 321; the entries between 1 and 2 are surrounded by parentheses.  Emeric Deutsch, Apr 06 2008
a(n) = Sum_{k=0..n1} k*A138770(n+1,k).  Emeric Deutsch, Apr 06 2008
a(n) is also the number of peaks in all permutations of {1,2,...,n+1}. Example: a(3)=16 because the permutations 1234, 4123, 3124, 4312, 2134, 4213, 3214, and 4321 have no peaks and each of the remaining 16 permutations of {1,2,3,4} has exactly one peak.  Emeric Deutsch, Jul 26 2009
a(n), for n>=2, is the number of (n+2)node tournaments that have exactly one triad. Proven by Kadane (1966), see link.  Ian R Harris, Sep 26 2022


REFERENCES

D. Daly and P. Vojtechovsky, Displacement of permutations, preprint, 2003.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..300
J. B. Kadane, Some equivalence classes in paired comparisons, The Annals of Mathematical Statistics, 37 (1966), 488494.


FORMULA

a(n) = A052571(n+2)/3 = 2*A005990(n).  Zerinvary Lajos, May 11 2007
a(n) = (n+3)! * Sum_{k=1..n} (k+1)!/(k+3)!, with offset 0.  Gary Detlefs, Aug 05 2010
E.g.f.: (x^3  3*x^2)/(3*(x1)^3).  Geoffrey Critzer, Mar 04 2013
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=2} 1/a(n) = (3/2)*(Ei(1)  gamma)  3*e + 27/4, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=2} (1)^n/a(n) = (3/2)*(gamma  Ei(1))  3/4, where Ei(1) = A099285. (End)


MATHEMATICA

nn=20; Drop[Range[0, nn]!CoefficientList[Series[ x^3/3/(1x)^2, {x, 0, nn}], x], 2] (* Geoffrey Critzer, Mar 04 2013 *)


PROG

(Magma) [(n^21)*Factorial(n)/3: n in [1..25]]; // Vincenzo Librandi, Oct 11 2011


CROSSREFS

Twice A005990.
Cf. A138770.
Cf. A001113, A001620, A091725, A099285.
Sequence in context: A026129 A026158 A025185 * A200820 A209361 A341925
Adjacent sequences: A090669 A090670 A090671 * A090673 A090674 A090675


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Dec 18 2003


STATUS

approved



