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A278460 a(n) is the number of size n Eulerian orientations in L1(2). 1
1, 2, 10, 66, 504, 4008, 32834, 275608, 2360480, 20556620, 181529552, 1621938126, 14637043030, 133227981608, 1221707669122, 11276264756890, 104678045392986, 976696485890714, 9154664125402120, 86159716204843096, 813905566526799242, 7714432262247963256, 73344196288917825312 (list; graph; refs; listen; history; text; internal format)
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

Cf. A277493.

Sequence in context: A278459 A278461 A027307 * A278462 A060206 A277493

Adjacent sequences:  A278457 A278458 A278459 * A278461 A278462 A278463

KEYWORD

nonn

AUTHOR

Gheorghe Coserea, Nov 22 2016

STATUS

approved

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Last modified September 15 22:10 EDT 2019. Contains 327088 sequences. (Running on oeis4.)