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A069726 Number of rooted planar bi-Eulerian maps with 2n edges. Bi-Eulerian: all its vertices and faces are of even valency. 3
1, 1, 6, 54, 594, 7371, 99144, 1412802, 21025818, 323686935, 5120138790, 82812679560, 1364498150904, 22839100002036, 387477144862128, 6651170184185802, 115346229450879978, 2018559015390399615, 35610482089433479410, 632770874050702595670, 11317118106279639106530 (list; graph; refs; listen; history; text; internal format)
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, 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

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 A081132 A158831

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

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Last modified November 19 03:53 EST 2017. Contains 294912 sequences.