A278462 a(n) is the number of size n Eulerian orientations in L2(2).

1, 2, 10, 66, 504, 4148, 35794, 319384, 2922472, 27273568, 258591700, 2483951402, 24121423518, 236417231844, 2335648036770, 23234567806722, 232536604918010, 2339766365713070, 23655013692855212, 240175288588925992, 2447961627258995826, 25037715732340569328, 256899046844874628732

Offset

0,2

Comments

For definition of the set L2(k), k>=1, see sec. 4, def. 6 in N. Bonichon et al. paper; in sec. 4.2, (21) gives the degree-6 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

G.f. y satisfies: 0 = 2*x^5*y^6 - x^4*(x+8)*y^5 - x^3*(3*x^2-16)*y^4 + x^2*(2*x+3)*(2*x-5)*y^3 - x*(2*x^2-7*x-7)*y^2 - (5*x+1)*y + 1.

Example

A(x) = 1 + 2*x + 10*x^2 + 66*x^3 + 504*x^4 + 4148*x^5 + ... is the g.f.

Mathematica

terms = 23;
A[_] = 0; Do[A[x_] = (1/(1 + 5x))(1 + (7x + 7x^2 - 2x^3) A[x]^2 + (-15x^2 - 4x^3 + 4x^4) A[x]^3 + (16x^3 - 3x^5) A[x]^4 + (-8x^4 - x^5) A[x]^5 + 2x^5 A[x]^6) + O[x]^terms // Normal, {terms}];
CoefficientList[A[x], x][[1 ;; terms]] (* Jean-François Alcover, Jul 25 2018 *)

Prog

(PARI)
x='x; y='y; Fxy = 2*x^5*y^6 - x^4*(x+8)*y^5 - x^3*(3*x^2-16)*y^4 + x^2*(2*x+3)*(2*x-5)*y^3 - x*(2*x^2-7*x-7)*y^2 - (5*x+1)*y + 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: A027307 A373325 A278460 * A060206 A277493 A361448

Adjacent sequences: A278459 A278460 A278461 * A278463 A278464 A278465

Keyword

nonn

Author

Gheorghe Coserea, Nov 23 2016

Status

approved