%I #41 Sep 26 2022 15:50:16

%S 0,2,16,120,960,8400,80640,846720,9676800,119750400,1596672000,

%T 22832409600,348713164800,5666588928000,97639686144000,

%U 1778437140480000,34145993097216000,689322235650048000,14597412049059840000,323575967087493120000,7493338185184051200000

%N a(n) = (n^2-1)*n!/3.

%C 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.

%C 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

%C a(n) = Sum_{k=0..n-1} k*A138770(n+1,k). - _Emeric Deutsch_, Apr 06 2008

%C 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

%C 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

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

%H Vincenzo Librandi, <a href="/A090672/b090672.txt">Table of n, a(n) for n = 1..300</a>

%H J. B. Kadane, <a href="https://doi.org/10.1214/aoms/1177699532">Some equivalence classes in paired comparisons</a>, The Annals of Mathematical Statistics, 37 (1966), 488-494.

%F a(n) = A052571(n+2)/3 = 2*A005990(n). - _Zerinvary Lajos_, May 11 2007

%F a(n) = (n+3)! * Sum_{k=1..n} (k+1)!/(k+3)!, with offset 0. - _Gary Detlefs_, Aug 05 2010

%F E.g.f.: (x^3 - 3*x^2)/(3*(x-1)^3). - _Geoffrey Critzer_, Mar 04 2013

%F From _Amiram Eldar_, May 14 2022: (Start)

%F Sum_{n>=2} 1/a(n) = (3/2)*(Ei(1) - gamma) - 3*e + 27/4, where Ei(1) = A091725, gamma = A001620, and e = A001113.

%F Sum_{n>=2} (-1)^n/a(n) = (3/2)*(gamma - Ei(-1)) - 3/4, where Ei(-1) = -A099285. (End)

%t nn=20;Drop[Range[0,nn]!CoefficientList[Series[ x^3/3/(1-x)^2,{x,0,nn}],x],2] (* _Geoffrey Critzer_, Mar 04 2013 *)

%o (Magma) [(n^2-1)*Factorial(n)/3: n in [1..25]]; // _Vincenzo Librandi_, Oct 11 2011

%Y Twice A005990.

%Y Cf. A138770.

%Y Cf. A001113, A001620, A091725, A099285.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Dec 18 2003