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%I #45 Sep 24 2018 17:45:12
%S 1,1,1,1,2,6,19,64,230,865,3364,13443,54938,228749,967628,4149024,
%T 18000758,78905518,349037335,1556494270,6991433386,31609302688,
%U 143755711433,657301771172,3020175361634,13939605844996,64604720622719
%N Number of rooted non-separable Eulerian planar maps with n edges.
%H Gheorghe Coserea, <a href="/A069728/b069728.txt">Table of n, a(n) for n = 0..500</a>
%H Daniel Birmajer, Juan B. Gil, Michael D. Weiner, <a href="https://arxiv.org/abs/1803.07727">A family of Bell transformations</a>, arXiv:1803.07727 [math.CO], 2018.
%H Hsien-Kuei Hwang, Mihyun Kang, Guan-Huei Duh, <a href="https://doi.org/10.4230/LIPIcs.AofA.2018.29">Asymptotic Expansions for Sub-Critical Lagrangean Forms</a>, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
%H V. A. Liskovets and T. R. S. Walsh, <a href="http://dx.doi.org/10.1016/j.disc.2003.09.015">Enumeration of Eulerian and unicursal planar maps</a>, Discr. Math., 282 (2004), 209-221.
%F G.f.: y = A(x) satisfies 0 = y^5 - y^4 - 12*x*y^3 + x*(16*x + 11)*y^2 - 8*x^2*y + x^2. - _Gheorghe Coserea_, Apr 12 2018
%F a(n) ~ 75*sqrt(65)/(4394*sqrt(Pi)) * n^(-5/2) * (128/25)^n. - _Gheorghe Coserea_, and _Vaclav Kotesovec_, Apr 12 2018
%F A(x) = 1 + serreverse((1+x)^2*(1+12*x-(1-4*x)^(3/2))/(2*(4*x+3)^2)); equivalently, it can be rewritten as A(x) = 1 + serreverse((y-1)*(y^2+y-1)^2/(y^3*(3*y-2)^2)), where y = A000108(x). - _Gheorghe Coserea_, Apr 14 2018
%e A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 64*x^7 + 230*x^8 + ...
%t Flatten[{1, Table[(Sum[(-1)^j*Binomial[2*n + j - 1, j] * Sum[(-1)^k*2^(n - j - k - 1)*Binomial[j, k] * Binomial[2*n, n - j - k - 1], {k, 0, Min[j, n - j - 1]}], {j, 0, n - 1}] - 2*Sum[(-1)^j*Binomial[2*n + j - 1, j] * Sum[(-1)^k*2^(n - j - k - 2) * Binomial[j, k]*Binomial[2*n, n - j - k - 2], {k, 0, Min[j, n - j - 2]}], {j, 0, n - 2}])/n, {n, 1, 30}]}] (* _Vaclav Kotesovec_, Apr 13 2018 *) (* In the article by Liskovets and Walsh, p. 218, E'ns(n), the factor -2*Sum[...] is missing. *)
%o (PARI)
%o seq(N) = {
%o my(x ='x+O('x^N), y=serreverse(x*(1+x/2-x^2/4)^2/(2*(1+x)^2)));
%o Vec(1+y/2-y^2/4);
%o };
%o seq(27) \\ _Gheorghe Coserea_, Apr 12 2018
%Y Cf. A000257.
%K nonn
%O 0,5
%A _Valery A. Liskovets_, Apr 07 2002
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