A278459 a(n) is the number of size n Eulerian orientations in L1(1).

1, 2, 10, 66, 466, 3458, 26650, 211458, 1716642, 14193282, 119115818, 1012129602, 8690293618, 75283480834, 657206992954, 5775816653314, 51060139789122, 453749755736834, 4051091496955978, 36319665678928962, 326850292861873426, 2951487063152265858, 26735348244277012570

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, (10) gives the quadratic equation for the g.f.

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..299

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*y^2 - (1-x)^2*y - x^2 - 2*x + 1.

Example

A(x) = 1 + 2*x + 10*x^2 + 66*x^3 + 466*x^4 + 3458*x^5 + ... is the g.f.

Mathematica

terms = 23;
A[_] = 0; Do[A[x_] = (1 - 2x - x^2 + 2x A[x]^2)/(1-x)^2 + 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*y^2 - (1-x)^2*y - x^2 - 2*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: A374568 A230050 A340467 * A278461 A372580 A027307

Adjacent sequences: A278456 A278457 A278458 * A278460 A278461 A278462

Keyword

nonn

Author

Gheorghe Coserea, Nov 22 2016

Status

approved