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%I #19 Jul 25 2018 08:35:53
%S 1,2,10,66,490,3898,32482,279882,2473362,22294194,204174842,
%T 1894462354,17771064186,168254374890,1605751354066,15431016952730,
%U 149191682979874,1450182228623458,14163576408858346,138924886089370082,1367918804901854218,13516246001650813338,133977227356098512834
%N a(n) is the number of size n Eulerian orientations in L2(1).
%C For definition of the set L2(k), k>=1, see sec. 4, def. 6 in N. Bonichon et al. paper; in sec. 4.2, (19) gives the cubic equation for the g.f.
%H Gheorghe Coserea, <a href="/A278461/b278461.txt">Table of n, a(n) for n = 0..300</a>
%H Nicolas Bonichon, Mireille Bousquet-Mélou, Paul Dorbec, Claire Pennarun, <a href="https://arxiv.org/abs/1610.09837">On the number of planar Eulerian orientations</a>, arXiv:1610.09837 [math.CO], 2016.
%F G.f. y satisfies: 0 = x^2*y^3 + x*(x-4)*y^2 + (2*x+1)*y - 1.
%e 1 + 2*x + 10*x^2 + 66*x^3 + 490*x^4 + 3898*x^5 + ...
%t terms = 23;
%t A[_] = 0; Do[A[x_] = (-1 - 4x A[x]^2 + x^2 A[x]^2 + x^2 A[x]^3)/(-1 - 2x) + O[x]^terms // Normal, {terms}];
%t CoefficientList[A[x], x][[1 ;; terms]] (* _Jean-François Alcover_, Jul 25 2018 *)
%o (PARI)
%o x='x; y='y; Fxy = x^2*y^3 + x*(x-4)*y^2 + (2*x+1)*y - 1;
%o seq(N) = {
%o my(y0 = 1 + O('x^N), y1=0);
%o for (k = 1, N,
%o y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);
%o if (y1 == y0, break()); y0 = y1);
%o Vec(y0);
%o };
%o seq(23)
%Y Cf. A277493.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Nov 23 2016
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