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A324314
Expansion of the generating function of quartic rooted planar Eulerian orientations, counted by vertices.
3
4, 35, 402, 5334, 77472, 1197459, 19371000, 324457562, 5585968752, 98334394470, 1763204413488, 32108212934100, 592453862089920, 11056844466818451, 208406532039094530, 3962553996669046002, 75926241110870890200, 1464873759398352892050, 28437685246012526646228, 555153133514289017625900
OFFSET
1,1
LINKS
Mireille Bousquet-Mélou, Andrew Elvey Price, Andrew Price, The generating function of planar Eulerian orientations, arXiv:1803.08265 [math.CO], 2018.
Mireille Bousquet-Mélou, Andrew Elvey Price, Paul Zinn-Justin, Eulerian orientations and the six-vertex model on planar map, arXiv:1902.07369 [math.CO], 2019. See Theorem 2.
FORMULA
G.f.: (1/(3t^2))*(t-3t^2-R(t)) where R(t) is A324313.
PROG
(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)); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Feb 21 2019
STATUS
approved