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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.

%I A069726 #65 May 19 2024 14:03:54
%S A069726 1,1,6,54,594,7371,99144,1412802,21025818,323686935,5120138790,
%T A069726 82812679560,1364498150904,22839100002036,387477144862128,
%U A069726 6651170184185802,115346229450879978,2018559015390399615,35610482089433479410,632770874050702595670,11317118106279639106530
%N A069726 Number of rooted planar bi-Eulerian maps with 2n edges. Bi-Eulerian: all its vertices and faces are of even valency.
%C A069726 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
%H A069726 G. C. Greubel, <a href="/A069726/b069726.txt">Table of n, a(n) for n = 0..650</a>
%H A069726 M. Bousquet-Mélou and A. Jehanne, <a href="http://arXiv.org/abs/math.CO/0504018">Polynomial equations with one catalytic variable, algebraic series and map enumeration</a>, arXiv:math/0504018 [math.CO], 2005.
%H A069726 M. Bousquet-Mélou and G. Schaeffer, <a href="http://dx.doi.org/10.1006/aama.1999.0673">Enumeration of planar constellations</a>, Adv. in Appl. Math. v.24 (2000), 337-368.
%H A069726 V. A. Kazakov, M. Staudacher and Th. Wynter, <a href="https://arxiv.org/abs/hep-th/9502132">Character expansion methods for matrix models of dually weighted graphs</a>, arxiv:hep-th/9502132, 1995; Commun. Math. Phys. 177 (1996), 451-468.
%H A069726 V. A. Liskovets and T. R. S. Walsh, <a href="http://dx.doi.org/10.1016/j.disc.2003.09.015">Enumeration of Eulerian and unicursal planar maps</a>, Discr. Math., 282 (2004), 209-221.
%F A069726 a(n) = 3^(n-1)*A000139(n).
%F A069726 a(0)=1, a(n) = 3^(n-1)*binomial(3n, n+1)/(n(2n+1)) for n >= 1.
%F A069726 G.f.: A(x) = (1 + 3*y - y^2)/3 where 3*x^2*y^3 - y + 1 = 0.
%F A069726 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.
%F A069726 a(n) ~ 2^(-2*n-1)*3^(4*n-1/2)/(sqrt(Pi)*n^(5/2)). - _Ilya Gutkovskiy_, Dec 04 2016
%F A069726 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
%F A069726 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
%p A069726 s := 4*(4-81*z)^(1/2): u := 36*I*z^(1/2): a := (s+u)^(1/3): b := (s-u)^(1/3):
%p A069726 gf := 1 + ((b+a)*s + 108*I*z^(1/2)*(b-a) - 32*(9*z+1))/(432*z):
%p A069726 simplify(series(gf, z, 22)): seq(coeff(%, z, n), n = 0..20);
%p A069726 # _Peter Luschny_, May 19 2024
%t A069726 Join[{1},Table[3^(n-1) Binomial[3n,n+1]/(n(2n+1)),{n,20}]] (* _Harvey P. Dale_, Oct 18 2013 *)
%o A069726 (PARI) A069726(n)=if(n,3^(n-1)*binomial(3*n,n+1)/n/(2*n+1),1)  \\ _M. F. Hasler_, Mar 26 2012
%Y A069726 Cf. A000139, A000257, A006402, A090372.
%K A069726 easy,nice,nonn
%O A069726 0,3
%A A069726 _Valery A. Liskovets_, Apr 07 2002
%E A069726 Entry revised by Editors of the OEIS, Mar 26 - 27 2012