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

0,3

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.

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000139, A000257, A006402, A090372.

Sequence in context: A201352 A186375 A231554 * A269477 A305602 A367471

Adjacent sequences: A069723 A069724 A069725 * A069727 A069728 A069729

Keyword

easy,nice,nonn

Author

Valery A. Liskovets, Apr 07 2002

Extensions

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

Status

approved