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%I #59 Feb 07 2022 02:24:18
%S 0,1,1,3,14,90,738,7364,86608,1173240,17990600,308055528,5826331440,
%T 120629547584,2713659864832,65909241461760,1718947213795328,
%U 47912968352783232,1421417290991105664,44717945211445216640,1487040748881346835200,52117255681017313721088
%N E.g.f.: REVERT(2*x/(1+exp(x))) = Sum_{n>=0} a(n)*x^n/n!.
%C a(n+1) is also number of ways to place n nonattacking composite pieces semi-rook + semi-bishop on an n X n board. Two semi-bishops (see A187235) do not attack each other if they are in the same northwest-southeast diagonal. Two semi-rooks do not attack each other if they are in the same column (see also semi-queens, A099152). - _Vaclav Kotesovec_, Dec 22 2011
%H Alois P. Heinz, <a href="/A088789/b088789.txt">Table of n, a(n) for n = 0..200</a>
%H V. Dotsenko, <a href="http://arxiv.org/abs/1110.0844">Pattern avoidance in labelled trees</a>, arXiv preprint arXiv:1110.0844 [math.CO], 2011.
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 716-719.
%H R. W. Whitty, <a href="https://dx.doi.org/10.1016/j.disc.2007.07.054">Rook polynomials on two-dimensional surfaces and graceful labellings of graphs</a>, Discrete Math., 308 (2008), 674-683.
%F E.g.f.: x/2 - LambertW(-x*exp(x/2)/2). - _Vladeta Jovovic_, Feb 12 2008
%F a(n) = (1/2^n)*Sum_{k=1..n} binomial(n,k)*k^(n-1)= A038049(n)/2^n, n>1. - _Vladeta Jovovic_, Feb 12 2008
%F Asymptotics: a(n)/(n-2)! ~ b * q^(n-1) * sqrt(n), where q = 1/(2*LambertW(1/exp(1))) = 1.795560738334311... is the root of the equation 2*q = exp(1+1/(2*q)) and b = 1/(2*LambertW(1/exp(1))) * sqrt((1+LambertW(1/exp(1)))/(2*Pi)) = 0.8099431005... - _Vaclav Kotesovec_, Dec 22 2011, updated Sep 25 2012
%p a:= n->coeff(series(x/2-LambertW(-1/2*x*exp(1/2*x)), x=0, n+1), x, n)*n!:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Aug 14 2008
%t Table[n!/2^n*Sum[2^j/j!*StirlingS2[n-1,n-j],{j,1,n}],{n,1,50}] (* _Vaclav Kotesovec_, Dec 25 2011 *)
%t With[{nmax = 50}, CoefficientList[Series[x/2 - LambertW[-x*Exp[x/2]/2], {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Nov 14 2017 *)
%o (PARI) a(n)=local(A); if(n<0,0,A=x+O(x^n);n!*polcoeff(serreverse(2*x/(1 + exp(x))), n))
%o (PARI) x='x+O('x^50); concat([0], Vec(serlaplace(x/2 - lambertw(-x*exp(x/2)/2)))) \\ _G. C. Greubel_, Nov 14 2017
%K nonn,nice
%O 0,4
%A _Paul D. Hanna_, Oct 15 2003
%E More terms from _Alois P. Heinz_, Aug 14 2008
%E Minor edits by _Vaclav Kotesovec_, Mar 31 2014
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