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%I #50 Jul 06 2020 10:40:37
%S 0,1,2,10,48,296,2080,16752,151424,1519744,16766208,201685760,
%T 2627316736,36847260672,553551644672,8868624615424,150943592939520,
%U 2719816264613888,51724646086475776,1035359388788391936,21759010038674358272,479027478333199482880
%N a(n) = n!*Sum_{i=0..n-1} (n-i)*(-2)^i/(i+1)!.
%C Arises from "Unfriendly Seating Arrangement" problem around a circular table.
%H Vincenzo Librandi, <a href="/A037256/b037256.txt">Table of n, a(n) for n = 0..200</a>
%H Philippe Flajolet, <a href="http://algo.inria.fr/libraries/autocomb/fatmen-html/fatmen1.html">A seating arrangement problem</a>
%H Philippe Flajolet, <a href="/A037256/a037256.pdf">A seating arrangement problem</a> [Cached copy]
%H Dave Freedman and Larry Shepp, <a href="http://dx.doi.org/10.1137/1004037">An unfriendly seating arrangement, Problem 62-3</a>, SIAM Review, Vol. 6 (1964), 180-182.
%F E.g.f.: (1-exp(-2*x))*(1-x)^(-2)/2.
%F a(n) = 2*(n-1)*a(n-1) - (n-4)*(n-1)*a(n-2) - 2*(n-2)*(n-1)*a(n-3). - _Vaclav Kotesovec_, Oct 08 2012
%F a(n) ~ (1-1/e^2)*n!*n/2. - _Vaclav Kotesovec_, Oct 08 2012
%F a(n) = (-2)^n + (n+1)!/2 - (3+n)*Gamma(1+n,-2)/(2*e^2). - _Benedict W. J. Irwin_, Jul 06 2020
%p f:=n->n!*add((n-i)*(-2)^i/(i+1)!,i=0..n-1);
%p [seq(f(n),n=0..50)]; # _N. J. A. Sloane_, Mar 29 2014
%t m = 19; CoefficientList[ Series[(1 - Exp[-2x])*(1/((1-x)^2*2)), {x, 0, m}], x]*Range[0, m]!
%t (* _Jean-François Alcover_, Jun 28 2011 *)
%t Flatten[{0,Table[n!*Sum[Sum[(-1)^j*2^j/(j+1)!,{j,0,k}],{k,0,n-1}],{n,1,20}]}] (* _Vaclav Kotesovec_, Oct 27 2012 *)
%o (PARI) x='x+O('x^66); concat([0],Vec(serlaplace((1-exp(-2*x))/(2*(1-x)^2)))) \\ _Joerg Arndt_, May 04 2013
%Y Cf. A095236, A239888, A239889.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
%E Entry revised by _N. J. A. Sloane_, Mar 29 2014
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