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A090672
a(n) = (n^2-1)*n!/3.
7
0, 2, 16, 120, 960, 8400, 80640, 846720, 9676800, 119750400, 1596672000, 22832409600, 348713164800, 5666588928000, 97639686144000, 1778437140480000, 34145993097216000, 689322235650048000, 14597412049059840000, 323575967087493120000, 7493338185184051200000
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..n-1} 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
J. B. Kadane, Some equivalence classes in paired comparisons, The Annals of Mathematical Statistics, 37 (1966), 488-494.
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*(x-1)^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/(1-x)^2, {x, 0, nn}], x], 2] (* Geoffrey Critzer, Mar 04 2013 *)
PROG
(Magma) [(n^2-1)*Factorial(n)/3: n in [1..25]]; // Vincenzo Librandi, Oct 11 2011
CROSSREFS
Twice A005990.
Cf. A138770.
Sequence in context: A026129 A026158 A025185 * A200820 A209361 A341925
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 18 2003
STATUS
approved