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%I #19 Nov 13 2023 10:47:09
%S 0,0,1,4,26,212,2108,24720,334072,5112544,87396728,1650607040,
%T 34132685120,767025716736,18612106195456,485013257865472,
%U 13509071081429888,400505695457942528,12592502771190979712,418524228123134068224,14661145374751901317888
%N Number of bounded regions in the Linial arrangement L_{n-1}.
%C Except for the initial 0, these are the absolute values of A349719. - _Ira M. Gessel_, Nov 01 2023
%H Richard P. Stanley, <a href="https://www.pnas.org/doi/epdf/10.1073/pnas.93.6.2620">Hyperplane arrangements, interval orders, and trees</a>, Proc. Natl. Acad. Sci. USA 93 (1996), 2620-2625.
%H Vasu Tewari, <a href="https://arxiv.org/abs/1604.06894">Gessel polynomials, rooks, and extended Linial arrangements</a>, arXiv preprint arXiv:1604.06894 [math.CO], 2016.
%F From _Ira M. Gessel_, Nov 01 2023: (Start)
%F a(n) = (1/2^n) * Sum_{k=0..n} (k-1)^(n-1) * binomial(n,k) for n>=2.
%F In the following generating functions we take a(1)=1 rather than a(1)=0.
%F E.g.f.: 1 + (1/2)*x/LambertW(-(1/2)*x*exp(x/2)).
%F E.g.f.: 1-1/B(x), where B(x) is the e.g.f. of A007889. See Corollary 4.2 of Stanley's paper. (End)
%F a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n) * 2^n * LambertW(exp(-1))^(n-1)). - _Vaclav Kotesovec_, Nov 13 2023
%Y Cf. A007889, A349719.
%K nonn
%O 1,4
%A _N. J. A. Sloane_, Mar 19 2017
%E More terms from _Ira M. Gessel_, Nov 01 2023
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