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
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
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 *)
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
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