A000257 -id:A000257 - cp's OEIS frontend

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

Valery A. Liskovets, Apr 07 2002

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

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.

Cf. A000139, A000257, A006402, A090372.

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

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

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

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

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

A318104 Number of genus 4 rooted hypermaps with n darts.

Original entry on oeis.org

8064, 579744, 23235300, 684173164, 16497874380, 344901105444, 6471056247920, 111480953909328, 1792031518697232, 27197316623478960, 393207192141924744, 5453210050430783640, 72949244341257096792, 945523594111460363208, 11918067649004916470640, 146538779626167833263888, 1762112462707129510538640

Offset: 9

Gheorghe Coserea, Nov 12 2018

nonn

Column k = 4 of A321710.

a(n) = 0 for n < 9. - N. J. A. Sloane, Dec 24 2018

			A(x) = 8064*x^9 + 579744*x^10 + 23235300*x^11 + 684173164*x^12 + ...

Gheorghe Coserea, Table of n, a(n) for n = 9..109
Mednykh, A.; Nedela, R. Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 6
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Cf. A000257, A118093, A214817, A214818, A321710.

y = (1 - Sqrt[1 - 8 x])/(4 x);
gf = -y (y-1)^9 (262 y^14 - 4716 y^13 + 78327 y^12 - 569134 y^11 + 3266910 y^10 - 12675726 y^9 + 37548087 y^8 - 82680972 y^7 + 137674842 y^6 - 170295272 y^5 + 152918277 y^4 - 94811622 y^3 + 37127810 y^2 - 7566846 y + 505869)/(4 (y-2)^17 (y+1)^13);
Drop[CoefficientList[gf + O[x]^26, x], 9] (* Jean-François Alcover, Feb 07 2019, from PARI *)

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
  Vec(-y*(y - 1)^9*(262*y^14 - 4716*y^13 + 78327*y^12 - 569134*y^11 + 3266910*y^10 - 12675726*y^9 + 37548087*y^8 - 82680972*y^7 + 137674842*y^6 - 170295272*y^5 + 152918277*y^4 - 94811622*y^3 + 37127810*y^2 - 7566846*y + 505869)/(4*(y - 2)^17*(y + 1)^13));
};
seq(17)

G.f.: -y*(y - 1)^9*(262*y^14 - 4716*y^13 + 78327*y^12 - 569134*y^11 + 3266910*y^10 - 12675726*y^9 + 37548087*y^8 - 82680972*y^7 + 137674842*y^6 - 170295272*y^5 + 152918277*y^4 - 94811622*y^3 + 37127810*y^2 - 7566846*y + 505869)/(4*(y - 2)^17*(y + 1)^13), where y = C(2*x), C being the g.f. for A000108.

A069728 Number of rooted non-separable Eulerian planar maps with n edges.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 19, 64, 230, 865, 3364, 13443, 54938, 228749, 967628, 4149024, 18000758, 78905518, 349037335, 1556494270, 6991433386, 31609302688, 143755711433, 657301771172, 3020175361634, 13939605844996, 64604720622719

Offset: 0

Valery A. Liskovets, Apr 07 2002

nonn

			A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 64*x^7 + 230*x^8 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..500
Daniel Birmajer, Juan B. Gil, Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Hsien-Kuei Hwang, Mihyun Kang, Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A000257.

Flatten[{1, Table[(Sum[(-1)^j*Binomial[2*n + j - 1, j] * Sum[(-1)^k*2^(n - j - k - 1)*Binomial[j, k] * Binomial[2*n, n - j - k - 1], {k, 0, Min[j, n - j - 1]}], {j, 0, n - 1}] - 2*Sum[(-1)^j*Binomial[2*n + j - 1, j] * Sum[(-1)^k*2^(n - j - k - 2) * Binomial[j, k]*Binomial[2*n, n - j - k - 2], {k, 0, Min[j, n - j - 2]}], {j, 0, n - 2}])/n, {n, 1, 30}]}] (* Vaclav Kotesovec, Apr 13 2018 *) (* In the article by Liskovets and Walsh, p. 218, E'ns(n), the factor -2*Sum[...] is missing. *)

seq(N) = {
  my(x ='x+O('x^N), y=serreverse(x*(1+x/2-x^2/4)^2/(2*(1+x)^2)));
  Vec(1+y/2-y^2/4);
};
seq(27) \\ Gheorghe Coserea, Apr 12 2018

G.f.: y = A(x) satisfies 0 = y^5 - y^4 - 12*x*y^3 + x*(16*x + 11)*y^2 - 8*x^2*y + x^2. - Gheorghe Coserea, Apr 12 2018
a(n) ~ 75*sqrt(65)/(4394*sqrt(Pi)) * n^(-5/2) * (128/25)^n. - Gheorghe Coserea, and Vaclav Kotesovec, Apr 12 2018
A(x) = 1 + serreverse((1+x)^2*(1+12*x-(1-4*x)^(3/2))/(2*(4*x+3)^2)); equivalently, it can be rewritten as A(x) = 1 + serreverse((y-1)*(y^2+y-1)^2/(y^3*(3*y-2)^2)), where y = A000108(x). - Gheorghe Coserea, Apr 14 2018

A085687 Expansion of g.f. 8/(1+sqrt(1-8*x))^3.

Original entry on oeis.org

1, 6, 36, 224, 1440, 9504, 64064, 439296, 3055104, 21498880, 152807424, 1095450624, 7911587840, 57511157760, 420459724800, 3089600348160, 22806128885760, 169033661153280, 1257467341701120, 9385880636620800, 70271680244613120, 527595313582571520

Offset: 0

N. J. A. Sloane, Jul 13 2003

nonn

a(n) is also the number of paths of length 2(n+1) in a binary tree between two vertices that are 2 steps apart. [David Koslicki (koslicki(AT)math.psu.edu), Nov 02 2010]

Cf. A002421, A000108, A000257.

CoefficientList[Series[8/(1 + Sqrt[1 - 8*x])^3, {x, 0, 21}], x] (* Amiram Eldar, Mar 24 2022 *)

a(n) = 6(n+1)*2^(n-2)*Cat(n+2)/(2n+3), where Cat(n)=A000108(n). - Ralf Stephan, Mar 11 2004
G.f.: c(2x)^3, where c(x) is the g.f. of A000108; a(n)=3(n+1)2^n*Cat(n+1)/(n+3); - Paul Barry, Dec 08 2004
a(n) = (n+1) * A000257(n+1). - F. Chapoton, Feb 26 2024
D-finite with recurrence: (n+3)*a(n) -2*(5*n+7)*a(n-1) +8*(2*n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2011
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 22/49 + 808*arcsin(1/(2*sqrt(2)))/(147*sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = 26/81 + 376*arcsinh(1/(2*sqrt(2)))/243. (End)

A090374 Number of rooted planar 4-constellations with n quadrangles: rooted planar maps with bicolored faces having n black quadrangular faces and an arbitrary number of white faces of degrees multiple to 4.

Original entry on oeis.org

1, 10, 160, 3200, 72960, 1813504, 47923200, 1325629440, 37991219200, 1120005652480, 33789432561664, 1039157228994560, 32480974549811200, 1029463445864448000, 33023079530417356800, 1070513886720329515008, 35026358912891580579840, 1155516042520241436098560

Offset: 1

Valery A. Liskovets, Dec 01 2003

M. Bousquet-Mélou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. v.24 (2000), 337-368.

Cf. A000257, A069726, A069729, A090373.

A090374 := proc(n)
    5*4^(n-1)*binomial(4*n, n)/((3*n+1)*(3*n+2))
end proc:
seq(A090374(n),n=1..40) ; # R. J. Mathar, Mar 29 2023

a[n_] := 5 2^(2n) (4n-1)! / ((n-1)! (3n+2)!);
Array[a, 18] (* Jean-François Alcover, Aug 28 2019 *)

vector(20, n, 5*4^(n-1)*binomial(4*n, n)/((3*n+1)*(3*n+2))) \\ Michel Marcus, Dec 11 2014

a(n) = 5*4^(n-1)*binomial(4*n, n)/((3*n+1)*(3*n+2)). - corrected by Michel Marcus, Dec 11 2014

D-finite with recurrence 3*n*(3*n+2)*(3*n+1)*a(n) -32*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Mar 29 2023

More terms from Michel Marcus, Dec 11 2014

A165546 Number of permutations of length n that avoid the patterns 3412 and 2413.

Original entry on oeis.org

1, 1, 2, 6, 22, 90, 395, 1823, 8741, 43193, 218704, 1129944, 5937728, 31656472, 170892498, 932625326, 5138618526, 28554124650, 159874462032, 901243508380, 5111776163584, 29155580007964, 167139065156182, 962618219420046

Offset: 0

Vincent Vatter, Sep 21 2009

a(n) is the number of permutations of length n avoiding the partially ordered pattern (POP) {3>1, 3>4, 1>2, 4>2} of length 4. That is, the number of length n permutations having no subsequences of length 4 in which the third element is the largest and the second element is the smallest. - Sergey Kitaev, Dec 11 2020

			There are 22 permutations of length 4 that avoid these two patterns, so a(4)=22.

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Darla Kremer and Wai Chee Shiu, Finite transition matrices for permutations avoiding pairs of length four patterns, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
Samuel Miner, Jay Pantone, Completing the Structural Analysis of the 2x4 Permutation Classes, arXiv:1802.00483 [math.CO], 2018.
Wikipedia, Permutation classes avoiding two patterns of length 4.

Cf. A000257.

nmax = 30; A[] = 0; Do[A[x] = x^4*A[x]^3 + (5*x - 11)*x^2*A[x]^2 + (3*x + 10)*x*A[x] - 9*x + 1 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Jul 05 2024 *)

A(x*B(x)) = (B(x)-1)/(x*B(x)^2), where B(x) is the o.g.f. for A000257 and A(x) is the o.g.f. for A165546. This can be proven using the generating function equation at the end of section 3 of Miner and Pantone's paper. - Michael D. Weiner, Jul 02 2024
a(n) ~ 2^(5*n + 8) / (81 * sqrt(Pi) * n^(5/2) * 5^(n + 1/2)). - Vaclav Kotesovec, Jul 05 2024
G.f.: (x - F(x))/x^2, where F(x) is the compositional inverse of x*B(x) and B(x) is the o.g.f. for A000257. This follows from Michael Weiner's comment above. - Alexander Burstein, Aug 02 2024

a(13)-a(14) (obtained by brute force enumeration) from Stephen DeSalvo, Sep 23 2015
a(15)-a(23) from David Bevan, Oct 03 2015
a(0)=1 prepended by Alois P. Heinz, Dec 09 2015

A181282 a(n) is the number of associate Rota-Baxter words in one idempotent generator x and one idempotent operator P of degree n. Such words are Rota-Baxter words that begin and/or end with x, and P is applied n times in the word.

Original entry on oeis.org

1, 3, 12, 60, 336, 2016, 12672, 82368, 549120, 3734016, 25798656, 180590592, 1278025728, 9128755200, 65727037440, 476521021440, 3475800391680, 25489202872320, 187815179059200, 1389832325038080, 10324468700282880

Offset: 0

William Sit (wyscc(AT)sci.ccny.cuny.edu), Oct 11 2010

nonn

			For n = 2, the a(2) = 12 associate Rota-Baxter words are: xP(xP(x)), xP(xP(x))x, P(xP(x))x, xP(P(x)x), xP(P(x)x)x, P(P(x)x)x, xP(xP(x)x), xP(xP(x)x)x, P(xP(x)x)x, xP(x)xP(x), xP(x)xP(x)x, P(x)xP(x)x.

Indranil Ghosh, Table of n, a(n) for n = 0..1000
L. Guo and W. Sit, Enumeration and generating functions of Rota-Baxter words, Math. Comp. Sci. (2010). In G. Regensburger, M. Rosenkranz, and W. Sit, eds., Algebraic and Algorithmic Aspects of Differential and Integral Operators (AADIOS), Sp. Issue, Math. C. Sc., 4 (2,3) (2010).

Cf. A000108, A003645, A025225, A000257.

[1] cat [3*2^(n-1)*Catalan(n): n in [1..40]]; // G. C. Greubel, Jan 04 2023

CoefficientList[Series[(3-4x-3Sqrt[1-8x])/(8x), {x,0,40}], x]
a[0] = 1; a[n_]:= 3*2^(n-1) CatalanNumber[n]; Table[a[n], {n,0,20}] (* Indranil Ghosh, Mar 05 2017 *)

a(n) = if(n==0, 1, 3*2^(n-1)*(binomial(2*n,n)/(n+1))); \\ Indranil Ghosh, Mar 05 2017

import math
f = math.factorial
def C(n,r): return f(n)/f(r)/f(n-r)
def A181282(n): return 1 if n==0 else 3*2**(n-1)*(C(2*n,n)/(n+1)) # Indranil Ghosh, Mar 05 2017

[3*2^(n-1)*catalan_number(n) -int(n==0)/2 for n in range(41)] # G. C. Greubel, Jan 04 2023

a(n) = 3*2^(n-1)*A000108(n).
G.f.: (3 - 4*t - 3*sqrt(1-8*t))/(8*t).
(n+1)*a(n) = 4*(2*n-1)*a(n-1). - R. J. Mathar, Jul 24 2012
a(n) = (n+2) * A000257(n). - F. Chapoton, Feb 26 2024

A298358 a(n) is the number of rooted 3-connected bicubic planar maps with 2n vertices.

Original entry on oeis.org

1, 0, 0, 1, 0, 3, 7, 15, 63, 168, 561, 1881, 6110, 21087, 72174, 250775, 883116, 3125910, 11174280, 40209852, 145590720, 530358095, 1941862860, 7144623447, 26403493545, 97971775008, 364903633215, 1363847131450, 5113975285788, 19233646581282

Offset: 1

Michael D. Weiner, Jan 17 2018

nonn

			A(x) = x + x^4 + 3*x^6 + 7*x^7 + 15*x^8 + 63*x^9 + 168*x^10 + 561*x^11 + ...

Gheorghe Coserea, Table of n, a(n) for n = 1..501
Daniel Birmajer, Juan B. Gil, Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Hsien-Kuei Hwang, Mihyun Kang, Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
W. T. Tutte, A census of planar maps, Canad. J. Math., 15(1963), 249-271.

Cf. A000257, A069728, A069729.

kmax = 30; b[0] = 1; b[n_] := 3*2^(n - 1)*CatalanNumber[n]/(n + 2);
G[x_] = Sum[b[k] x^k, {k, 0, kmax}];
A[_] = 1;
Do[A[x_] = G[x/(1 + A[x] + O[x]^k)^3] - 1 // Normal, {k, 1, kmax + 1}];
CoefficientList[A[x], x][[2 ;; -2]] (* Jean-François Alcover, Jun 19 2018 *)

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x), g=(1+4*y-y^2)/4);
  Vec(subst(g-1, 'x, serreverse(x*g^3)));
};
seq(30) \\ Gheorghe Coserea, Apr 11 2018

G.f.: A(x) = G(x/(1+A(x))^3)-1 where A(x*(G(x))^3) = G(x)-1 and G(x) = g.f. for A000257.
From Gheorghe Coserea, Apr 16 2018: (Start)
y = A(x)/x satisfies: 0 = x^6*y^7 + 6*x^5*y^6 + 15*x^4*y^5 + 4*x^3*(5 - 3*x)*y^4 + x^2*(15 - 37*x)*y^3 + x*(16*x^2 - 39*x + 6)*y^2 + (24*x^2 - 15*x + 1)*y + (9*x - 1).
A(x) = serreverse((1+x)^3*(1 + 12*x - (1-4*x)^(3/2))/(2*(4*x+3)^2)); equivalently, it can be rewritten as A(x) = serreverse((y-1)*(y^2+y-1)^3/(y^5*(3*y-2)^2)), where y = A000108(x). (End)
a(n) ~ 3 * 2^(9*n-1) / (sqrt(Pi) * 17^(5/2) * 5^(3*n - 5/2) * n^(5/2)). - Gheorghe Coserea and Vaclav Kotesovec, Apr 16 2018

A003584 Unicursal (i.e., possessing an Eulerian path) planar rooted maps with n edges.

Original entry on oeis.org

1, 2, 9, 52, 336, 2304, 16368, 118976, 878592, 6562816, 49447424, 375072768, 2860343296, 21909012480, 168425533440, 1298753372160, 10041201131520, 77809145610240, 604138825973760, 4698956908462080, 36604934482821120

Offset: 0

Valery A. Liskovets

Unicursal (in a broad sense) means that no more than two vertices are of odd valency (that is maps possessing an Eulerian path or cycle).

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A000257, A005470, A069720.

a[n_] := 2^(n-1)*(3*Binomial[2*n, n]/((n+1)*(n+2))+Binomial[2*n-1, n]); a[0]=1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 11 2014 *)

a(n) = A000257(n) + A069720(n).

More terms from Valery A. Liskovets, Apr 07 2002

A101477 Square array T(n,k), read by antidiagonals: number of labeled trees, with increments of labels along edges constrained to +-1, with n nodes that have no label greater than k.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 2, 7, 12, 1, 2, 8, 31, 56, 1, 2, 8, 39, 156, 288, 1, 2, 8, 40, 211, 851, 1584, 1, 2, 8, 40, 223, 1219, 4909, 9152, 1, 2, 8, 40, 224, 1327, 7371, 29506, 54912, 1, 2, 8, 40, 224, 1343, 8250, 46099, 183043, 339456, 1, 2, 8, 40, 224, 1344, 8427, 52938, 295915, 1164387, 2149888

Offset: 0

Ralf Stephan, Jan 21 2005

			1, 1, 3, 12,  56,  288, 1584,  9152,  54912,  339456, ...
1, 2, 7, 31, 156,  851, 4909, 29506, 183043, 1164387, ...
1, 2, 8, 39, 211, 1219, 7371, 46099, 295915, 1939395, ...
1, 2, 8, 40, 223, 1327, 8250, 52938, 347941, 2330532, ...
1, 2, 8, 40, 224, 1343, 8427, 54625, 362833, 2456261, ...
1, 2, 8, 40, 224, 1344, 8447, 54887, 365688, 2484384, ...
1, 2, 8, 40, 224, 1344, 8448, 54911, 366051, 2488831, ...
1, 2, 8, 40, 224, 1344, 8448, 54912, 366079, 2489311, ...
1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489343, ...
1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489344, ...

M. Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005.

Rows converge to A052701. First row is A000257.

Cf. A052701, A101478.

nmax = 11;
b[x_] = Sum[2^(n - 1)*(2*n - 2)!/(n - 1)!/n! x^n, {n, 1, nmax}];
c[x_] = 0; Do[c[x_] = x*(1 + c[x])^4/(1 + c[x]^2) + O[x]^nmax, {nmax}];
a[n_, t_] := a[n, t] = b[t]*(1 - c[t]^(n + 1))*(1 - c[t]^(n + 5))/((1 - c[t]^(n + 2))*(1 - c[t]^(n + 4)));
T[n_, k_] := SeriesCoefficient[a[n, t], {t, 0, k}];
Table[T[n - k, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 25 2018 *)

G.f. of k-th row: A(t)=B(t)*(1-C(t)^(k+1))*(1-C(t)^(k+5))/[(1-C(t)^(k+2))*(1-C(t)^(k+4))], with tB(t) the g.f. of A052701 and C(t) the g.f. of A101478.

cp's OEIS Frontend

A069726 Number of rooted planar bi-Eulerian maps with 2n edges. Bi-Eulerian: all its vertices and faces are of even valency.

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A318104 Number of genus 4 rooted hypermaps with n darts.

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A069728 Number of rooted non-separable Eulerian planar maps with n edges.

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A085687 Expansion of g.f. 8/(1+sqrt(1-8*x))^3.

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A090374 Number of rooted planar 4-constellations with n quadrangles: rooted planar maps with bicolored faces having n black quadrangular faces and an arbitrary number of white faces of degrees multiple to 4.

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A165546 Number of permutations of length n that avoid the patterns 3412 and 2413.

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A181282 a(n) is the number of associate Rota-Baxter words in one idempotent generator x and one idempotent operator P of degree n. Such words are Rota-Baxter words that begin and/or end with x, and P is applied n times in the word.

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