OFFSET
0,2
COMMENTS
For definition of the set L1(k), k>=1, see sec. 3, def. 1 in N. Bonichon et al. paper; in sec. 3.2, (12) gives the quartic equation for the g.f.
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 0..300
Nicolas Bonichon, Mireille Bousquet-Mélou, Paul Dorbec, Claire Pennarun, On the number of planar Eulerian orientations, arXiv:1610.09837 [math.CO], 2016.
FORMULA
0 = 8*x^3*y^4 - 4*x^2*(3*x^3+4*x^2-6*x+3)*y^3 + 2*x*(3*x^5-12*x^4-10*x^3+14*x^2-10*x+3)*y^2 + (x-1)*(11*x^5-10*x^4-6*x^3-3*x^2-x+1)*y + (x-1)*(5*x^5-4*x^4+6*x^3-7*x^2+5*x-1), where y is the g.f.
EXAMPLE
A(x) = 1 + 2*x + 10*x^2 + 66*x^3 + 504*x^4 + 4008*x^5 + ... is the g.f.
MATHEMATICA
y[_] = 0; Do[y[x_] = -((1 + 6x (-1 + y[x]^2) + x^6 (5 + 6y[x]^2) - 4x^2 (-3 + 5y[x]^2 + 3y[x]^3) - 3x^5 (3 + 8y[x]^2 + 4y[x]^3) - 2x^4 (-5 + 10 y[x]^2 + 8y[x]^3) + x^3 (-13 + 28y[x]^2 + 24y[x]^3 + 8y[x]^4))/(-1 + 2x + 2x^2 + 3x^3 + 4x^4 - 21x^5 + 11x^6)) + O[x]^n // Normal, {n, 23}];
CoefficientList[y[x], x] (* Jean-François Alcover, Oct 23 2018 *)
PROG
(PARI)
x='x; y='y;
Fxy = 8*x^3*y^4 - 4*x^2*(3*x^3+4*x^2-6*x+3)*y^3 + 2*x*(3*x^5-12*x^4-10*x^3+14*x^2-10*x+3)*y^2 + (x-1)*(11*x^5-10*x^4-6*x^3-3*x^2-x+1)*y + (x-1)*(5*x^5-4*x^4+6*x^3-7*x^2+5*x-1);
seq(N) = {
my(y0 = 1 + O('x^N), y1=0);
for (k = 1, N,
y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);
if (y1 == y0, break()); y0 = y1);
Vec(y0);
};
seq(23)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Nov 22 2016
STATUS
approved