|
| |
|
|
A069726
|
|
Number of rooted planar bi-Eulerian maps with 2n edges. Bi-Eulerian: all its vertices and faces are of even valency.
|
|
2
| |
|
|
1, 1, 6, 54, 594, 7371, 99144, 1412802, 21025818, 323686935, 5120138790, 8281267956, 136449815090, 2283910000203, 38747714486212
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| a(n)=3^(n-1)*S'(n+1) where S'(n) denotes the number of rooted non-separable planar maps with n edges (the sequence A006402).
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 Liskovets (liskov(AT)im.bas-net.by), Dec 01 2003
|
|
|
REFERENCES
| V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.
|
|
|
LINKS
| V. A. Kazakov, M. Staudacher and Th. Wynter, Character expansion methods for matrix models of dually weighted graphs, Commun. Math. Phys. 177 (1996), 451-468.
M. Bousquet-Melou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. v.24 (2000), 337-368.
M. Bousqet-Melou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration
|
|
|
FORMULA
| a(n)=3^(n-1)*binomial(3n, n+1)/[n(2n+1)] G.f.: f(x)=(1+3y-y^2)/3 where 3x^2y^3-y+1=0.
G.f. satisfies A(z) = 1-47z+3z^2+3z(22-9z)A+9z(9z-2)A^2-81z^2A^3.
|
|
|
CROSSREFS
| Cf. A000257, A006402.
Cf. A090372.
Sequence in context: A109576 A201352 A186375 * A081132 A158831 A034001
Adjacent sequences: A069723 A069724 A069725 * A069727 A069728 A069729
|
|
|
KEYWORD
| easy,nice,nonn
|
|
|
AUTHOR
| Valery A. Liskovets (liskov(AT)im.bas-net.by), Apr 07 2002
|
| |
|
|
