A069728 Number of rooted non-separable Eulerian planar maps with n edges.

1, 1, 1, 1, 2, 6, 19, 64, 230, 865, 3364, 13443, 54938, 228749, 967628, 4149024, 18000758, 78905518, 349037335, 1556494270, 6991433386, 31609302688, 143755711433, 657301771172, 3020175361634, 13939605844996, 64604720622719

Offset: 0

Author

Valery A. Liskovets, Apr 07 2002

Keywords

• nonn

Examples

			A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 64*x^7 + 230*x^8 + ...
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..500
• Daniel Birmajer, Juan B. Gil, Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
• Hsien-Kuei Hwang, Mihyun Kang, Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
• V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Crossrefs

Cf. A000257.

Programs

Mathematica

Flatten[{1, Table[(Sum[(-1)^j*Binomial[2*n + j - 1, j] * Sum[(-1)^k*2^(n - j - k - 1)*Binomial[j, k] * Binomial[2*n, n - j - k - 1], {k, 0, Min[j, n - j - 1]}], {j, 0, n - 1}] - 2*Sum[(-1)^j*Binomial[2*n + j - 1, j] * Sum[(-1)^k*2^(n - j - k - 2) * Binomial[j, k]*Binomial[2*n, n - j - k - 2], {k, 0, Min[j, n - j - 2]}], {j, 0, n - 2}])/n, {n, 1, 30}]}] (* Vaclav Kotesovec, Apr 13 2018 *) (* In the article by Liskovets and Walsh, p. 218, E'ns(n), the factor -2*Sum[...] is missing. *)

PARI

seq(N) = {
  my(x ='x+O('x^N), y=serreverse(x*(1+x/2-x^2/4)^2/(2*(1+x)^2)));
  Vec(1+y/2-y^2/4);
};
seq(27) \\ Gheorghe Coserea, Apr 12 2018

Formula

G.f.: y = A(x) satisfies 0 = y^5 - y^4 - 12*x*y^3 + x*(16*x + 11)*y^2 - 8*x^2*y + x^2. - Gheorghe Coserea, Apr 12 2018

a(n) ~ 75*sqrt(65)/(4394*sqrt(Pi)) * n^(-5/2) * (128/25)^n. - Gheorghe Coserea, and Vaclav Kotesovec, Apr 12 2018

A(x) = 1 + serreverse((1+x)^2*(1+12*x-(1-4*x)^(3/2))/(2*(4*x+3)^2)); equivalently, it can be rewritten as A(x) = 1 + serreverse((y-1)*(y^2+y-1)^2/(y^3*(3*y-2)^2)), where y = A000108(x). - Gheorghe Coserea, Apr 14 2018