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%I #25 Aug 21 2019 15:06:36
%S 1,1,3,13,71,486,3982,38081,416145,5116222,69888746,1050168417,
%T 17214678241,305703953660,5846391071172,119794781201881,
%U 2618283427770737,60802908515558346,1495049717728972990,38803241993010963977,1060124286228724147641,30411290829335509535632
%N Expansion of e.g.f. (2*exp(x)-2*x-x^2)/(2-2*x-x^2).
%C Number of totally ordered partitions on an n-element set where each non-minimal class contains at most 2 elements.
%C Convention a(0) = 1.
%H Michael De Vlieger, <a href="/A307005/b307005.txt">Table of n, a(n) for n = 0..427</a>
%H Jimmy Devillet, <a href="http://hdl.handle.net/10993/39776">On the single-peakedness property</a>, International summer school "Preferences, decisions and games" (Sorbonne Université, Paris, 2019).
%H J. Devillet, J.-L. Marichal, and B. Teheux <a href="https://arxiv.org/abs/1811.11113">Classifications of quasitrivial semigroups</a>, arXiv:1811.11113 [math.RA], 2018.
%F Recurrence: a(1) = 1, a(2) = 3, a(n+2) = 1 + (n+2)*a(n+1) + (1/2)*(n+2)*(n+1)*a(n).
%F a(n) = Sum_{i=0..n} (n!/(n + 1 - i)!)*((sqrt(3)/3)*((1 + sqrt(3))/2)^i - (sqrt(3)/3)*((1 - sqrt(3))/2)^i).
%t Nest[Append[#1, 1 + #2 #1[[-1]] + #2 (#2 - 1) #1[[-2]]/2 ] & @@ {#, Length@ #} &, {1, 1, 3}, 19] (* _Michael De Vlieger_, Apr 21 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace((2*exp(x)-2*x-x^2)/(2-2*x-x^2))) \\ _Felix Fröhlich_, Mar 19 2019
%Y Cf. A307006.
%K nonn,easy
%O 0,3
%A _J. Devillet_, Mar 19 2019
%E More terms from _Michel Marcus_, Apr 20 2019