A238357 -id:A238357 - cp's OEIS frontend

A000168 a(n) = 23^n(2n)!/(n!(n+2)!).

1, 2, 9, 54, 378, 2916, 24057, 208494, 1876446, 17399772, 165297834, 1602117468, 15792300756, 157923007560, 1598970451545, 16365932856990, 169114639522230, 1762352559231660, 18504701871932430, 195621134074714260, 2080697516976506220, 22254416920705240440, 239234981897581334730, 2583737804493878415084

Offset: 0

N. J. A. Sloane

Number of rooted planar maps with n edges. - Don Knuth, Nov 24 2013
Number of rooted 4-regular planar maps with n vertices.
Also, number of doodles with n crossings, irrespective of the number of loops.
From Karol A. Penson, Sep 02 2010: (Start)
Integral representation as n-th moment of a positive function on the (0,12) segment of the x axis. This representation is unique as it is the solution of the Hausdorff moment problem.
a(n) = Integral_{x=0..12} ((x^n*(4/9)*(1 - x/12)^(3/2)) / (Pi*sqrt(x/3))).  (End)
Also, the number of distinct underlying shapes of closed normal linear lambda terms of a given size, where the shape of a lambda term abstracts away from its variable binding. [N. Zeilberger, 2015] - N. J. A. Sloane, Sep 18 2016
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018
Number of well-labeled trees (Bona, 2015). - N. J. A. Sloane, Dec 25 2018

			G.f. = 1 + 2*x + 9*x^2 + 54*x^3 + 378*x^4 + 2916*x^5 + 24057*x^6 + 208494*x^7 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 319, 353.
E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.
V. A. Liskovets, A census of nonisomorphic planar maps, in Algebraic Methods in Graph Theory, Vol. II, ed. L. Lovasz and V. T. Sos, North-Holland, 1981.
V. A. Liskovets, Enumeration of nonisomorphic planar maps, Selecta Math. Sovietica, 4 (No. 4, 1985), 303-323.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..925 [Terms 0 to 100 computed by T. D. Noe; terms 101 to 925 by G. C. Greubel, Jan 15 2017]
Marie Albenque and Dominique Poulalhon, A Generic Method for Bijections between Blossoming Trees and Planar Maps, Electron. J. Combin., 22 (2015), #P2.38.
J.-L. Baril, R. Genestier, A. Giorgetti, and A. Petrossian, Rooted planar maps modulo some patterns, Preprint 2016.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Valentin Bonzom, Guillaume Chapuy, Maciej Dolega, Enumeration of non-oriented maps via integrability, Alg. Combin. 5 (6) (2022) p 1363-1390, A.1.
M. Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005.
M. Bousquet-Mélou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration, arXiv:math/0504018 [math.CO], 2005.
Sean R. Carrell and Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.
R. Cori and B. Vauquelin, Planar maps are well labeled trees, Canad. J. Math., 33 (1981), 1023-1042.
S. R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 516
A. Giorgetti, R. Genestier, and V. Senni, Software Engineering and Enumerative Combinatorics, slides from a talk at MAP 2014.
Hsien-Kuei Hwang, Mihyun Kang, and 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.
C. Kassel, On combinatorial zeta functions, Slides from a talk, Potsdam, 2015.
Sergey Kitaev, Anna de Mier, and Marc Noy, On the number of self-dual rooted maps, European J. Combin. 35 (2014), 377--387. MR3090510. See Eq. (1). - _N. J. A. Sloane_, May 19 2014
Evgeniy Krasko and Alexander Omelchenko, Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus, the projective plane and the Klein bottle. Sensed maps on the torus, Discrete Mathematics (2019) Vol. 342, Issue 2, 584-599. Also arXiv:1709.03225 [math.CO].
V. A. Liskovets, Enumeration of nonisomorphic planar maps, Journal of Graph Theory, Volume 5, Issue 1, pages 115-117, Spring 1981.
Valery A. Liskovets, A reductive technique for enumerating non-isomorphic planar maps, Discrete Math. 156 (1996), no. 1-3, 197--217. MR1405018 (97f:05087) - From _N. J. A. Sloane_, Jun 03 2012
R. C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, 3 (1967), 103-121.
R. C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, 3 (1967), 103-121. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
C. Reutenauer and M. Robado, On an algebraicity theorem of Kontsevich, FPSAC 2012, Nagoya, Japan DMTCS proc. AR, 2012, 241-248. - From _N. J. A. Sloane_, Dec 23 2012
G. Schaeffer and P. Zinn-Justin, On the asymptotic number of plane curves and alternating knots, arXiv:math-ph/0304034, 2003-2004.
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.
Noam Zeilberger, Counting isomorphism classes of beta-normal linear lambda terms, arXiv:1509.07596 [cs.LO], 2015.
Noam Zeilberger, Towards a mathematical science of programming, Preprint 2015.
Noam Zeilberger, Linear lambda terms as invariants of rooted trivalent maps, arXiv preprint arXiv:1512.06751 [cs.LO], 2015.
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030, March 2018 (A revised version of a 2017 conference paper)
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Part 2, Rutgers Experimental Math Seminar, Sep 13 2018.
Noam Zeilberger and Alain Giorgetti, A correspondence between rooted planar maps and normal planar lambda terms, arXiv:1408.5028 [cs.LO], 2014-2015; Logical Methods in Computer Science, vol. 11 (3:22), 2015, pp. 1-39.
Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
First row of array A101486.
Cf. A005470.
Rooted maps with n edges of genus g for 0 <= g <= 10: this sequence, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.

[(2*Catalan(n)*3^n)/(n+2): n in [1..30]]; // Vincenzo Librandi, Sep 04 2014

Maple
```
A000168:=n->2*3^n*(2*n)!/(n!*(n+2)!);
```

Table[(2*3^n*(2n)!)/(n!(n+2)!),{n,0,20}] (* Harvey P. Dale, Jul 25 2011 *)
a[ n_] := If[ n < 0, 0, 2 3^n (2 n)!/(n! (n + 2)!)] (* Michael Somos, Nov 25 2013 *)
a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/2, 1, 3, 12 x], {x, 0, n}] (* Michael Somos, Nov 25 2013 *)

{a(n) = if( n<0, 0, 2 * 3^n * (2*n)! / (n! * (n+2)!))}; /* Michael Somos, Nov 25 2013 */

G.f. A(z) satisfies A(z) = 1 - 16*z + 18*z*A(z) - 27*z^2*A(z)^2.
G.f.: F(1/2,1;3;12x). - Paul Barry, Feb 04 2009
a(n) = 2*3^n*A000108(n)/(n+2). - Paul Barry, Feb 04 2009
D-finite with recurrence: (n + 1) a(n) = (12 n - 18) a(n - 1). - Simon Plouffe, Feb 09 2012
G.f.: 1/54*(-1+18*x+(-(12*x-1)^3)^(1/2))/x^2. - Simon Plouffe, Feb 09 2012
0 = a(n)*(+144*a(n+1) - 42*a(n+2)) + a(n+1)*(+18*a(n+1) + a(n+2)) if n>=0. - Michael Somos, Jan 31 2014
a(n) ~ 2*(12^n)/((n^2+3*n)*sqrt(Pi*n)). - Peter Luschny, Nov 25 2015
E.g.f.: exp(6*x)*(12*x*BesselI(0,6*x) - (1 + 12*x)*BesselI(1,6*x))/(9*x). - Ilya Gutkovskiy, Feb 01 2017
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 1887/1331 + 3240*arccosec(2*sqrt(3))/(1331*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 1563/2197 - 3240*arccosech(2*sqrt(3))/(2197*sqrt(13)). (End)

More terms from Joerg Arndt, Feb 26 2014

A006300 Number of rooted maps with n edges on torus.

Original entry on oeis.org

1, 20, 307, 4280, 56914, 736568, 9370183, 117822512, 1469283166, 18210135416, 224636864830, 2760899996816, 33833099832484, 413610917006000, 5046403030066927, 61468359153954656, 747672504476150374, 9083423595292949240, 110239596847544663002, 1336700736225591436496, 16195256987701502444284

Offset: 2

N. J. A. Sloane

E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

T. D. Noe, Table of n, a(n) for n = 2..100
D. Arquès, Relations fonctionnelles et dénombrement des cartes pointées sur le tore, J. Combin. Theory Ser. B, 43 (1987), 253-274.
E. A. Bender, E. R. Canfield and R. W. Robinson, The enumeration of maps on the torus and the projective plane, Canad. Math. Bull., 31 (1988), 257-271.
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).
S. R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, preprint (submitted to J. Combin. Th. B).
A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, Discrete Mathematics, Volume 310, Issue 3, 6 February 2010, pp. 518-526.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
T. R. S. Walsh, Counting maps on doughnuts, Theoretical Computer Science, vol. 502, pp. 4-15, (September-2013).

Column k=1 of A238396.
Cf. A007137, A006386.
Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, this sequence, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.

R:=sqrt(1-12*x): seq(coeff(convert(series((R-1)^2/(12*R^2*(R+2)),x,50),polynom),x,n),n=2..25); (Pab Ter)

Drop[With[{c=Sqrt[1-12x]},CoefficientList[Series[(c-1)^2/(12c^2 (c+2)), {x,0,30}],x]],2] (* Harvey P. Dale, Jun 14 2011 *)

A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A006300_ser(N) = my(y = A005159_ser(N+1)); y*(y-1)^2/(3*(y-2)^2*(y+2));
Vec(A006300_ser(21)) \\ Gheorghe Coserea, Jun 02 2017

G.f.: (R-1)^2/(12*R^2*(R+2)) where R=sqrt(1-12*x); a(n) is asymptotic to 12^n/24. - Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005
a(n) = Sum_{k=0..n-2} 2^(n-3-k)*(3^(n-1)-3^k)*binomial(n+k,k). - Ruperto Corso, Dec 18 2011
D-finite with recurrence: n*a(n) +22*(-n+1)*a(n-1) +4*(22*n-65)*a(n-2) +96*(5*n-4)*a(n-3) +576*(-2*n+7)*a(n-4)=0. - R. J. Mathar, Feb 20 2020

Bender et al. give 20 terms.
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005
More terms from Joerg Arndt, Feb 26 2014

A269919 Triangle read by rows: T(n,g) is the number of rooted maps with n edges on an orientable surface of genus g.

Original entry on oeis.org

1, 2, 9, 1, 54, 20, 378, 307, 21, 2916, 4280, 966, 24057, 56914, 27954, 1485, 208494, 736568, 650076, 113256, 1876446, 9370183, 13271982, 5008230, 225225, 17399772, 117822512, 248371380, 167808024, 24635754, 165297834, 1469283166, 4366441128

Offset: 0

Gheorghe Coserea, Mar 07 2016

Row n contains floor((n+2)/2) terms.

Equivalently, T(n,g) is the number of rooted bipartite quadrangulations with n faces of an orientable surface of genus g.

			Triangle starts:
n\g    [0]          [1]          [2]          [3]          [4]
[0]    1;
[1]    2;
[2]    9,           1;
[3]    54,          20;
[4]    378,         307,         21;
[5]    2916,        4280,        966;
[6]    24057,       56914,       27954,       1485;
[7]    208494,      736568,      650076,      113256;
[8]    1876446,     9370183,     13271982,    5008230,     225225;
[9]    17399772,    117822512,   248371380,   167808024,   24635754;
[10]   ...

Gheorghe Coserea, Rows n = 0..200, flattened
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Columns g=0-10 give: A000168, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.
Cf. A269418, A269419.
Same as A238396 except for the zeros.

T[0, 0] = 1; T[n_, g_] /; g<0 || g>n/2 = 0; T[n_, g_] := T[n, g] = ((4n-2)/ 3 T[n-1, g] + (2n-3)(2n-2)(2n-1)/12 T[n-2, g-1] + 1/2 Sum[(2k-1)(2(n-k)- 1) T[k-1, i] T[n-k-1, g-i], {k, 1, n-1}, {i, 0, g}])/((n+1)/6);
Table[T[n, g], {n, 0, 10}, {g, 0, n/2}] // Flatten (* Jean-François Alcover, Jul 20 2018 *)

N = 9; gmax(n) = n\2;
Q = matrix(N+1, N+1);
Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
Qset(n, g, v) = { Q[n+1, g+1] = v };
Quadric({x=1}) = {
  Qset(0, 0, x);
  for (n = 1, N, for (g = 0, gmax(n),
    my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
       t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
       t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
       (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
    Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
};
Quadric();
concat(vector(N+1, n, vector(1 + gmax(n-1), g, Qget(n-1, g-1))))

(n+1)/6 * T(n, g) = (4*n-2)/3 * T(n-1, g) + (2*n-3)*(2*n-2)*(2*n-1)/12 * T(n-2, g-1) + 1/2 * Sum_{k=1..n-1} Sum_{i=0..g} (2*k-1) * (2*(n-k)-1) * T(k-1, i) * T(n-k-1, g-i) for all n >= 1 and 0 <= g <= n/2, with the initial conditions T(0,0) = 1 and T(n,g) = 0 for g < 0 or g > n/2.

For column g, as n goes to infinity we have T(n,g) ~ t(g) * n^(5*(g-1)/2) * 12^n, where t(g) = (A269418(g)/A269419(g)) / (2^(g-2) * gamma((5*g-1)/2)) and gamma is the Gamma function.

A006301 Number of rooted genus-2 maps with n edges.

Original entry on oeis.org

0, 0, 0, 0, 21, 966, 27954, 650076, 13271982, 248371380, 4366441128, 73231116024, 1183803697278, 18579191525700, 284601154513452, 4272100949982600, 63034617139799916, 916440476048146056, 13154166812674577412, 186700695099591735024, 2623742783421329300190, 36548087103760045010148, 505099724454854883618924

Offset: 0

N. J. A. Sloane and Simon Plouffe

nonn

E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

T. D. Noe, Table of n, a(n) for n=0..30 (from Mednykh and Nedela)
E. A. Bender and E. R. Canfield, The number of rooted maps on an orientable surface, J. Combin. Theory, B 53 (1991), 293-299.
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).
S. R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.

Column k=2 of A238396.

Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, this sequence, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 20 2018 *)

A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A006301_ser(N) = {
  my(y=A005159_ser(N+1));
  -y*(y-1)^4*(4*y^4 - 16*y^3 + 153*y^2 - 148*y + 196)/(9*(y-2)^7*(y+2)^4);
};
concat([0,0,0,0], Vec(A006301_ser(19))) \\ Gheorghe Coserea, Jun 02 2017

More terms from Joerg Arndt, Feb 26 2014

A104742 Number of rooted maps of (orientable) genus 3 containing n edges.

Original entry on oeis.org

1485, 113256, 5008230, 167808024, 4721384790, 117593590752, 2675326679856, 56740864304592, 1137757854901806, 21789659909226960, 401602392805341924, 7165100439281414160, 124314235272290304540, 2105172926498512761984, 34899691847703927826500, 567797719808735191344672, 9084445205688065541367710

Offset: 6

Valery A. Liskovets, Mar 22 2005

nonn

T. D. Noe, Table of n, a(n) for n = 6..30 (from Mednykh and Nedela)
E. A. Bender and E. R. Canfield, The number of rooted maps on an orientable surface, J. Combin. Theory, B 53 (1991), 293-299.
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).
S. R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, preprint (submitted to J. Combin. Th. B).

Column k=3 of A238396.
Cf. A104596 (unrooted maps).
Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, A006301, this sequence, A215402, A238355, A238356, A238357, A238358, A238359, A238360.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 3];
Table[a[n], {n, 6, 30}] (* Jean-François Alcover, Jul 20 2018 *)

A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A104742_ser(N) = {
  my(y=A005159_ser(N+1));
  y*(y-1)^6*(460*y^8 - 3680*y^7 + 63055*y^6 - 198110*y^5 + 835954*y^4 - 1408808*y^3 + 1986832*y^2 - 1462400*y + 547552)/(81*(y-2)^12*(y+2)^7)
};
Vec(A104742_ser(17))  \\ Gheorghe Coserea, Jun 02 2017

A215402 Number of rooted maps of (orientable) genus 4 containing n edges.

Original entry on oeis.org

225225, 24635754, 1495900107, 66519597474, 2416610807964, 75981252764664, 2141204115631518, 55352670009315660, 1334226671709010578, 30347730709395639732, 657304672067357799042, 13652607304062788395788, 273469313030628783700080, 5306599156694095573465824, 100128328831437989131706976, 1842794650155970906232185656

Offset: 8

Alain Giorgetti, Aug 09 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 8..500
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).
Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
Alexander Mednykh, Alain Giorgetti, Enumeration of genus four maps by number of edges, Ars Mathematica Contemporanea 4 (2011), 351--361.

Row sums of A269924.
Column g=4 of A269919.
Cf. A215019 (unrooted sensed maps), A297880 (unrooted unsensed maps).
Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, A006301, A104742, this sequence, A238355, A238356, A238357, A238358, A238359, A238360.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 4];
Table[a[n], {n, 8, 30}] (* Jean-François Alcover, Jul 20 2018 *)

A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A215402_ser(N) = {
  my(y=A005159_ser(N+1));
  -y*(y-1)^8*(15812*y^12 - 189744*y^11 + 4708549*y^10 - 24892936*y^9 + 173908449*y^8 - 567987942*y^7 + 1743939189*y^6 - 3485359548*y^5 + 5448471852*y^4 - 6051484928*y^3 + 4633500336*y^2 - 2228416192*y + 517976128)/(81*(y-2)^17*(y+2)^10);
};
Vec(A215402_ser(16)) \\ Gheorghe Coserea, Jun 02 2017

More terms from Joerg Arndt, Feb 26 2014

A238355 Number of rooted maps of genus 5 containing n edges.

Original entry on oeis.org

59520825, 8608033980, 672868675017, 37680386599440, 1692352190653740, 64755027944420400, 2190839204960030106, 67194704604610557072, 1901727022434216910002, 50322107898515282999256, 1257582616997225194094310, 29916524874047762719113408, 681758763997451748190036272, 14960113428664295584816860864

Offset: 10

Joerg Arndt, Feb 26 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 10..500
Sean R. Carrell and Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014)
Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.

Row sums of A269925.
Column g=5 of A269919.
Cf. A239918 (unrooted sensed), A348798 (unrooted unsensed)
Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, A006301, A104742, A215402, this sequence, A238356, A238357, A238358, A238359, A238360.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 5];
Table[a[n], {n, 10, 30}] (* Jean-François Alcover, Jul 20 2018 *)

PARI
```
\\ see A238396
```

A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A238355_ser(N) = {
  my(y=A005159_ser(N+1));
  y*(y-1)^10*(3149956*y^16 - 50399296*y^15 + 1641189689*y^14 - 12178227918*y^13 + 118643174857*y^12 - 572499071300*y^11 + 2690451915197*y^10 - 8657342508522*y^9 + 23652302179098*y^8 - 49891059998872*y^7 + 84432024838000*y^6 - 112355956173344*y^5 + 115338024848256*y^4 - 88846084908160*y^3 + 48488699816960*y^2 - 16837415717888*y + 2841312026112)/(243*(y-2)^22*(y+2)^13);
};
Vec(A238355_ser(14)) \\ Gheorghe Coserea, Jun 02 2017

A238356 Number of rooted maps of genus 6 containing n edges.

Original entry on oeis.org

24325703325, 4416286056750, 425671555397220, 28948474436455224, 1558252224413413380, 70639804918689629112, 2802850363447807024080, 99911395098598706576856, 3259947795252652107008514, 98729808377337068918681196, 2805432194025270702468165744

Offset: 12

Joerg Arndt, Feb 26 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 12..500
Sean R. Carrell and Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).
Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.

Column g=6 of A269919.
Cf. A239919 (unrooted sensed), A348798 (unrooted unsensed).
Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, A006301, A104742, A215402, A238355, this sequence, A238357, A238358, A238359, A238360.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 6];
Table[a[n], {n, 12, 30}] (* Jean-François Alcover, Jul 20 2018 *)

PARI
```
\\ see A238396
```

A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A238356_ser(N) = {
  my(y=A005159_ser(N+1));
  -y*(y-1)^12*(3091382412*y^20 - 61827648240*y^19 + 2494741456179*y^18 - 23821030780564*y^17 + 297709107215018*y^16 - 1898397937026724*y^15 + 11996625283021532*y^14 - 53079600835119544*y^13 + 206468965657569764*y^12 - 637634273350412392*y^11 + 1660605297373850222*y^10 - 3573247507645221112*y^9 + 6390852378647917144*y^8 - 9449999309170921856*y^7 + 11435897504002339264*y^6 - 11175919884930946304*y^5 + 8621441033651120896*y^4 - 5068129528843341824*y^3 + 2141653827725309440*y^2 - 581932716954417152*y + 76958488611567616)/(2187*(y-2)^27*(y+2)^16);
};
Vec(A238356_ser(11)) \\ Gheorghe Coserea, Jun 02 2017

cp's OEIS Frontend

A000168 a(n) = 2*3^n*(2*n)!/(n!*(n+2)!).

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A006300 Number of rooted maps with n edges on torus.

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A269919 Triangle read by rows: T(n,g) is the number of rooted maps with n edges on an orientable surface of genus g.

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A006301 Number of rooted genus-2 maps with n edges.

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A104742 Number of rooted maps of (orientable) genus 3 containing n edges.

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A215402 Number of rooted maps of (orientable) genus 4 containing n edges.

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A238355 Number of rooted maps of genus 5 containing n edges.

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A000168 a(n) = 23^n(2n)!/(n!(n+2)!).