A069726 Number of rooted planar bi-Eulerian maps with 2n edges. Bi-Eulerian: all its vertices and faces are of even valency.

1, 1, 6, 54, 594, 7371, 99144, 1412802, 21025818, 323686935, 5120138790, 82812679560, 1364498150904, 22839100002036, 387477144862128, 6651170184185802, 115346229450879978, 2018559015390399615, 35610482089433479410, 632770874050702595670, 11317118106279639106530

Offset: 0

Author

Valery A. Liskovets, Apr 07 2002

Keywords

• easy
• nice
• nonn

Comments

Also counts rooted planar 3-constellations with n triangles: rooted planar maps with bicolored faces having n black triangular faces and an arbitrary number of white faces of degrees multiple to 3. - Valery A. Liskovets, Dec 01 2003

Links

• G. C. Greubel, Table of n, a(n) for n = 0..650
• M. Bousquet-Mélou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration, arXiv:math/0504018 [math.CO], 2005.
• M. Bousquet-Mélou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. v.24 (2000), 337-368.
• V. A. Kazakov, M. Staudacher and Th. Wynter, Character expansion methods for matrix models of dually weighted graphs, arxiv:hep-th/9502132, 1995; Commun. Math. Phys. 177 (1996), 451-468.
• V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Crossrefs

Cf. A000139, A000257, A006402, A090372.

Programs

Maple

s := 4*(4-81*z)^(1/2): u := 36*I*z^(1/2): a := (s+u)^(1/3): b := (s-u)^(1/3):
gf := 1 + ((b+a)*s + 108*I*z^(1/2)*(b-a) - 32*(9*z+1))/(432*z):
simplify(series(gf, z, 22)): seq(coeff(%, z, n), n = 0..20);
# Peter Luschny, May 19 2024

Mathematica

Join[{1},Table[3^(n-1) Binomial[3n,n+1]/(n(2n+1)),{n,20}]] (* Harvey P. Dale, Oct 18 2013 *)

PARI

A069726(n)=if(n,3^(n-1)*binomial(3*n,n+1)/n/(2*n+1),1)  \\ M. F. Hasler, Mar 26 2012

Formula

a(n) = 3^(n-1)*A000139(n).

a(0)=1, a(n) = 3^(n-1)*binomial(3n, n+1)/(n(2n+1)) for n >= 1.

G.f.: A(x) = (1 + 3*y - y^2)/3 where 3*x^2*y^3 - y + 1 = 0.

G.f. satisfies A(z) = 1 -47*z +3*z^2 +3*z*(22-9*z)*A(z) +9*z*(9*z-2)*A(z)^2 -81*z^2*A(z)^3.

a(n) ~ 2^(-2*n-1)*3^(4*n-1/2)/(sqrt(Pi)*n^(5/2)). - Ilya Gutkovskiy, Dec 04 2016

D-finite with recurrence 2*(n+1)*(2*n+1)*a(n) -9*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Mar 29 2023

G.f. 1/3 - 2/(27*z) + sqrt(4 - 81*z)*((sqrt(4 - 81*z)/2 + 9*i*sqrt(z)/2)^(1/3) + (sqrt(4 - 81*z)/2 - 9*i*sqrt(z)/2)^(1/3))/(54*z) - (((sqrt(4 - 81*z)/2 + 9*i*sqrt(z)/2)^(1/3) - (sqrt(4 - 81*z)/2 - 9*i*sqrt(z)/2)^(1/3))*i)/(2*sqrt(z)), where i = sqrt(-1). - Karol A. Penson, May 19 2024

Extensions

Entry revised by Editors of the OEIS, Mar 26 - 27 2012