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%I #7 Feb 22 2019 06:03:52
%S 4,35,402,5334,77472,1197459,19371000,324457562,5585968752,
%T 98334394470,1763204413488,32108212934100,592453862089920,
%U 11056844466818451,208406532039094530,3962553996669046002,75926241110870890200,1464873759398352892050,28437685246012526646228,555153133514289017625900
%N Expansion of the generating function of quartic rooted planar Eulerian orientations, counted by vertices.
%H Mireille Bousquet-Mélou, Andrew Elvey Price, Andrew Price, <a href="https://arxiv.org/abs/1803.08265">The generating function of planar Eulerian orientations</a>, arXiv:1803.08265 [math.CO], 2018.
%H Mireille Bousquet-Mélou, Andrew Elvey Price, Paul Zinn-Justin, <a href="https://arxiv.org/abs/1902.07369">Eulerian orientations and the six-vertex model on planar map</a>, arXiv:1902.07369 [math.CO], 2019. See Theorem 2.
%F G.f.: (1/(3t^2))*(t-3t^2-R(t)) where R(t) is A324313.
%o (PARI) lista(nn) = {nn += 2; my(v = vector(nn), R, P, c, r, s); kill(y); for (n=1, nn, v[n] = y; R = sum(k=1, n, v[k]*t^k); P = sum(k=0, n, binomial(2*k,k)*binomial(3*k,k)/(k+1)*R^(k+1)); c = polcoef(P, n, t); r = polcoef(c, 0, y); s = polcoef(c, 1, y); if (n==1, v[n] = (1-r)/s, v[n] = -r/s);); R = sum(k=1, #v, v[k]*t^k); vector(nn-2, k, polcoef((t - 3*t^2 - R)/(3*t^2), k, t));}
%Y Cf. A324311, A324312, A324313.
%K nonn
%O 1,1
%A _Michel Marcus_, Feb 21 2019
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