A342981 -id:A342981 - cp's OEIS frontend

A000260 Number of rooted simplicial 3-polytopes with n+3 nodes; or rooted 3-connected triangulations with 2n+2 faces; or rooted 3-connected trivalent maps with 2n+2 vertices.

1, 1, 3, 13, 68, 399, 2530, 16965, 118668, 857956, 6369883, 48336171, 373537388, 2931682810, 23317105140, 187606350645, 1524813969276, 12504654858828, 103367824774012, 860593023907540, 7211115497448720, 60776550501588855

Offset: 0

N. J. A. Sloane

Number of rooted loopless planar maps with n edges. E.g., there are a(2)=3 loopless planar maps with 2 edges: two rooted paths (.-.-.) and one digon (.=.). - Valery A. Liskovets, Sep 25 2003
Number of intervals (i.e., ordered pairs (x,y) such that x<=y) in the Tamari lattice (rotation lattice of binary trees) of size n (see Pallo and Chapoton references). - Ralf Stephan, May 08 2007, Jean Pallo (Jean.Pallo(AT)u-bourgogne.fr), Sep 11 2007
Number of rooted triangulations of type [n, 0] (see Brown paper eq (4.8)). - Michel Marcus, Jun 23 2013
Equivalently, number of rooted bridgeless planar maps with n edges. - Noam Zeilberger, Oct 06 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 uniquely sorted permutations of [2n+1] that avoid the pattern 231. Also the number of uniquely sorted permutations of [2n+1] that avoid 132. - Colin Defant, Jun 13 2019
The sequence 1,3,13,68,... appears naturally in integral geometry, namely in the algebra of unitarily invariant valuations on complex space forms. - Andreas Bernig, Feb 02 2020

			G.f. = 1 + x + 3*x^2 + 13*x^3 + 68*x^4 + 399*x^5 + 2530*x^6 + 16965*x^7 + ...

C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.
Handbook of Combinatorics, North-Holland '95, p. 891.
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).
W. T. Tutte, The enumerative theory of planar maps, in A Survey of Combinatorial Theory (J. N. Srivastava et al. eds.), pp. 437-448, North-Holland, Amsterdam, 1973.

T. D. Noe, Table of n, a(n) for n = 0..200
E. A. Bender and N. C. Wormald, The number of loopless planar maps, Discr. Math. 54:2 (1985), 235-237.
Andreas Bernig, Unitarily invariant valuations and Tutte's sequence, arXiv:2001.03372 [math.DG], 2020.
Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, Diagonals of permutahedra and associahedra, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See pp. 10-11.
Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
T. Daniel Brennan, Christian Ferko, and Savdeep Sethi, A Non-Abelian Analogue of DBI from $T \overline{T}$, arXiv:1912.12389 [hep-th], 2019. See also SciPost Phys. (2020) Vol. 8, 052.
Enrica Duchi and Corentin Henriet, A bijection between Tamari intervals and extended fighting fish, arXiv:2206.04375 [math.CO], 2022.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
Frédéric Chapoton, Sur le nombre d'intervalles dans les treillis de Tamari, arXiv:math/0602368 [math.CO], 2006.
Grégory Chatel, Vincent Pilaud, and Viviane Pons, The weak order on integer posets, arXiv:1701.07995 [math.CO], 2017.
Grégory Chatel and Viviane Pons, Counting smaller elements in the Tamari and m-Tamari lattices, arXiv preprint arXiv:1311.3922 [math.CO], 2013.
François David, A model of random surfaces with nontrivial critical behavior, Nuclear Physics B, v. 257 (1985), 543-576.
Colin Defant, Catalan Intervals and Uniquely Sorted Permutations, arXiv:1904.02627 [math.CO], 2019.
C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)
Wenjie Fang, Planar triangulations, bridgeless planar maps and Tamari intervals, arXiv:1611.07922 [math.CO], 2016.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
C. Germain and J. Pallo, The number of coverings in four Catalan lattices, Intern. J. Computer Math., Vol. 61 (1996) pp. 19-28. (See p. 27.)
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.
Zhaoxiang Li and Yanpei Liu, Chromatic sums of general maps on the sphere and the projective plane, Discr. Math. 307 (2007), 78-87.
Michaël Moortgat, The Tamari order for D^3 and derivability in semi-associative Lambek-Grishin Calculus, 15th Workshop: Computational Logic and Applications (CLA 2020).
K. A. Penson, K. Górska, A. Horzela, and G. H. E. Duchamp, Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution, arXiv:2209.06574 [math.CO], 2022.
Viviane Pons, Combinatorics of the Permutahedra, Associahedra, and Friends, arXiv:2310.12687 [math.CO], 2023. See p. 37.
W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38. See Eq. 5.12.
William T. Tutte, On the enumeration of convex polyhedra, J. Combin. Theory Ser. B 28 (1980), no. 2, 105-126. MR0572468 (81j:05073).
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, eq. (6).
Noam Zeilberger, A sequent calculus for the Tamari order, arXiv:1701.02917 [cs.LO], 2017.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030 [math.LO], 2018-2019 (A revised version of a 2017 conference paper).
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. See also Part 2.

Row sums of A342981.
Column 0 of A146305 and A341856; Column 2 of A255918.
Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Cf. A000256, A006013, A027836, A069271, A263191, A290326.

[Binomial(4*n+1, n+1)-9*Binomial(4*n+1, n-1): n in [0..25]]; // Vincenzo Librandi, Nov 24 2016

A000260 := proc(n)
    2*(4*n+1)!/((n+1)!*(3*n+2)!) ;
end proc:

Table[Binomial[4n+1,n+1]-9*Binomial[4n+1,n-1],{n,0,25}] (* Harvey P. Dale, Aug 23 2011 *)
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/2, 3/4, 1, 5/4}, {4/3, 5/3, 2}, 256/27 x], {x, 0, n}]; (* Michael Somos, Dec 23 2014 *)
terms = 22; G[] = 0; Do[G[x] = 1+x*G[x]^4 + O[x]^terms, terms];
CoefficientList[(2-G[x])*G[x]^2, x] (* Jean-François Alcover, Jan 13 2018, after Mark van Hoeij *)

{a(n) = if( n<0, 0, 2 * (4*n + 1)! / ((n + 1)! * (3*n + 2)!))}; /* Michael Somos, Sep 07 2005 */

{a(n) = binomial( 4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2))}; /* Michael Somos, Mar 28 2012 */

def a(n):
    n = ZZ(n)
    return (4*n + 2).binomial(n + 1) // ((2*n + 1) * (3*n + 2))
# F. Chapoton, Aug 06 2015

a(n) = 2*(4*n+1)! / ((n+1)!*(3*n+2)!) = binomial(4*n+1, n+1) - 9*binomial(4*n+1, n-1).
G.f.: (2-g)*g^2 where g = 1 + x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 10 2011
G.f.: hypergeom([1,1/2,3/4,5/4],[2,4/3,5/3],256*x/27) = 1 + 120*x/(Q(0)-120*x); Q(k) = 8*x*(2*k+1)*(4*k+3)*(4*k+5) + 3*(k+2)*(3*k+4)*(3*k+5) - 24*x*(k+2)*(2*k+3)*(3*k+4)*(3*k+5)*(4*k+7)*(4*k+9)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 25 2011
a(n) = binomial(4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2)). - Michael Somos, Mar 28 2012
a(n) * (n+1) = A069271(n). - Michael Somos, Mar 28 2012
0 = F(a(n), a(n+1), ..., a(n+8)) for all n in Z where a(-1) = 3/4 and F() is a polynomial of degree 2 with integer coefficients and 29 monomials. - Michael Somos, Dec 23 2014
D-finite with recurrence: 3*(3*n+2)*(3*n+1)*(n+1)*a(n) - 8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Oct 21 2015
a(n) = Sum_{k=1..A000108(n)} k * A263191(n,k). - Alois P. Heinz, Nov 16 2015
a(n) ~ 2^(8*n+7/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n+5/2)). - Vaclav Kotesovec, Feb 26 2016
E.g.f.: 3F3(1/2,3/4,5/4; 4/3,5/3,2; 256*x/27). - Ilya Gutkovskiy, Feb 01 2017
From Gheorghe Coserea, Aug 17 2017: (Start)
G.f. y(x) satisfies:
0 = x^3*y^4 + 3*x^2*y^3 + x*(8*x+3)*y^2 - (20*x-1)*y + 16*x-1.
0 = x*(256*x - 27)*deriv(y,x) - 8*x^2*y^3 - 25*x*y^2 + 4*(24*x-11)*y + 44.
(End)
From Karol A. Penson, Apr 06 2022: (Start)
a(n) = Integral_{x=0...256/27} x^n*W(x), where
W(x) = (sqrt(2)/Pi)*(h1(x) - h2(x) + h3(x)) and
h1(x) = 3F2([-6/12,-2/12,  2/12], [ 3/12,  9/12], (27*x)/256)/((x/2)^(1/2));
h2(x) = 3F2([-3/12, 1/12,  5/12], [ 6/12, 15/12], (27*x)/256)/(x^(1/4));
h3(x) = 3F2([ 3/12, 7/12, 11/12], [18/12, 21/12], (27*x)/256)/(x^(-1/4)*32).
This integral representation is unique as the solution of n-th Hausdorff power moment of the function W. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0 and for x > 0 is monotonically decreasing to zero at x = 256/27. (End)
a(n) = (1/27^n) * Product_{1 <= i <= j <= 3*n} (3*i + j + 3)/(3*i + j - 1). Cf. A006013. - Peter Bala, Feb 21 2023

Edited by F. Chapoton, Feb 03 2011

A006419 a(n) = 2^(2n+1) - C(2n+3,n+1) + C(2*n+1,n).

Original entry on oeis.org

0, 1, 7, 37, 176, 794, 3473, 14893, 63004, 263950, 1097790, 4540386, 18696432, 76717268, 313889477, 1281220733, 5219170052, 21224674118, 86188320962, 349550141078, 1416102710912, 5731427140268, 23177285611082, 93655986978002, 378195990166136, 1526289367335244

Offset: 0

N. J. A. Sloane

nonn

Number of rooted isthmusless planar maps with n+1 faces and 2 vertices. - Dan Drake, Aug 08 2005
a(n) = total area below all Dyck (n+1)-paths and above the lowest possible Dyck path, namely, UDUD...UD (taking upsteps of unit length). For example, the areas below the 5 Dyck 3-paths UUUDDD, UUDUDD, UDUUDD, UUDDUD, UDUDUD are 3,2,1,1,0 respectively, yielding a(2)=3+2+1+1+0=7. - David Callan, Jul 03 2006
Convolution of A000245 and A000302 (powers of 4).- Philippe Deléham, Jun 02 2013

			G.f. = x + 7*x^2 + 37*x^3 + 176*x^4 + 794*x^5 + 3473*x^6 + 14893*x^7 + 63004*x^8 + ...

D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..1660
Jason Bandlow and Kendra Killpatrick, An area-to-inv bijection between Dyck paths and 312-avoiding permutations,Electron. J. Combin. 8 (2001), no. 1, Research Paper 40, 16 pp.
Miklós Bóna, Surprising Symmetries in Objects Counted by Catalan Numbers, Electronic J. Combin., 19 (2012), #P62, eq. (5).
Pudwell, Lara; Scholten, Connor; Schrock, Tyler; Serrato, Alexa Noncontiguous pattern containment in binary trees, ISRN Comb. 2014, Article ID 316535, 8 p. (2014), Table 1.
R. P. Stanley, F. Zanello, The Catalan case of Armstrong's conjecture on core partitions, arXiv preprint arXiv:1312.4352 [math.CO], 2013.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259.

A diagonal of A342981.

Cf. A386612, A386614, A386616, A386617.

f := n->2^(2*n+1)-binomial(2*n+3,n+1)+binomial(2*n+1,n); seq(f(n), n=0..30);

Table[2^(2 n + 1) - Binomial[2 n + 3, n + 1] +
Binomial[2 n + 1, n], {n, 0, 30}] (* Wesley Ivan Hurt, Mar 30 2014 *)

a(n):=sum(binomial(2*(n+1),n-k-1),k,0,n); /* Vladimir Kruchinin, Oct 23 2016 */

a(n+1) = Sum_{k=0..n} (n-k)*A000108(n-k)*A001700(k). - Philippe Deléham, Jan 25 2004
G.f.: c(x)^3*x/(1-4x) where c(x) = g.f. for the Catalan numbers A000108. - Philippe Deléham, Jun 02 2013
a(n) = Integral_{x=0..4} x^n*W(x)*dx, n >= 0, is the integral representation as n-th moment of a signed weight function W(x), where W(x) = W_a(x) + W_c(x), with W_a(x) = 2*Dirac(x-4), which is the discrete (atomic) part, and W_c(x) = (1/(2*Pi))*(1-x)*sqrt(x/(4-x)) is the continuous part of W(x): W_c(0) = W_c(1) = 0, W_c(x) > 0 for x < 1, lim_{x->4} W_c(x) = -oo. - Karol A. Penson, Jul 31 2013 [edited by Michel Marcus, Mar 14 2020]
(n+2)*a(n) + (-9*n-10)*a(n-1) + 2*(12*n+1)*a(n-2) + 8*(-2*n+3)*a(n-3) = 0. - R. J. Mathar, Mar 30 2014
a(n) = Sum_{k=0..n} binomial(2*(n+1), n-k-1). - Vladimir Kruchinin, Oct 23 2016
0 = a(n)*(+256*a(n+1) - 992*a(n+2) + 520*a(n+3) - 72*a(n+4)) + a(n+1)*(+224*a(n+1) + 344*a(n+2) - 398*a(n+3) + 70*a(n+4)) + a(n+2)*(+6*a(n+2) + 59*a(n+3) - 17*a(n+4)) + a(n+3)*(-a(n+3) + a(n+4)), for all n >= 0. - Michael Somos, Oct 23 2016
a(n) = [x^n] x/((1 - 2*x)*(1 - x)^(n+3)). - Ilya Gutkovskiy, Oct 25 2017
From Seiichi Manyama, Jul 29 2025: (Start)
a(n) = Sum_{k=0..n-1} binomial(2*k+1+l,k) * binomial(2*n-2*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 2^(n-k-1) * binomial(n+k+2,k). (End)

A343092 Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without isthmuses, n >= 2, k = 1..n-1.

Original entry on oeis.org

1, 4, 10, 10, 79, 70, 20, 340, 900, 420, 35, 1071, 5846, 7885, 2310, 56, 2772, 26320, 71372, 59080, 12012, 84, 6258, 93436, 431739, 706068, 398846, 60060, 120, 12768, 280120, 2000280, 5494896, 6052840, 2499096, 291720, 165, 24090, 739420, 7643265, 32055391, 58677420, 46759630, 14805705, 1385670

Offset: 2

Andrew Howroyd, Apr 04 2021

The number of vertices is n - k.

Column k is a polynomial of degree 3*k. This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.

			Triangle begins:
   1;
   4,   10;
  10,   79,    70;
  20,  340,   900,    420;
  35, 1071,  5846,   7885,   2310;
  56, 2772, 26320,  71372,  59080,  12012;
  84, 6258, 93436, 431739, 706068, 398846, 60060;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..1276
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, Table VId.

Columns 1..2 are A000292, A006469.
Diagonals are A002802, A006425, A006426, A006427.
Row sums are A343093.
Cf. A269921, A342981, A342989, A343090.

\\ Needs F from A342989.
G(n,m,y,z)={my(p=F(n,m,y,z)); subst(p, x, serreverse(x*p^2))}
H(n, g=1)={my(q=G(n, g, 'y, 'z)-x, v=Vec(polcoef(sqrt(serreverse(x/q^2)/x), g, 'y))); [Vecrev(t) | t<-v]}
{ my(T=H(10)); for(n=1, #T, print(T[n])) }

A342980 Triangle read by rows: T(n,k) is the number of rooted loopless planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 8, 1, 0, 0, 1, 20, 20, 1, 0, 0, 1, 38, 131, 38, 1, 0, 0, 1, 63, 469, 469, 63, 1, 0, 0, 1, 96, 1262, 3008, 1262, 96, 1, 0, 0, 1, 138, 2862, 12843, 12843, 2862, 138, 1, 0, 0, 1, 190, 5780, 42602, 83088, 42602, 5780, 190, 1, 0

Offset: 0

Andrew Howroyd, Apr 01 2021

The number of vertices is n + 2 - k.

For k >= 2, columns k without the initial zero term is a polynomial of degree 3*(k-2). This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.

			Triangle begins:
  1;
  0, 0;
  0, 1,   0;
  0, 1,   1,    0;
  0, 1,   8,    1,     0;
  0, 1,  20,   20,     1,     0;
  0, 1,  38,  131,    38,     1,    0;
  0, 1,  63,  469,   469,    63,    1,   0;
  0, 1,  96, 1262,  3008,  1262,   96,   1, 0;
  0, 1, 138, 2862, 12843, 12843, 2862, 138, 1, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, Table VIa.

Columns (and diagonals) are A006416, A006417, A006418.
Row sums are A099553(n+1).
Cf. A082680, A269920, A342981, A342985, A343090.

G[m_, y_] := Sum[x^n*Sum[(n + k - 1)!*(2*n - k)!*y^k/(k!*(n + 1 - k)!*(2*k - 1)!*(2*n - 2*k + 1)!), {k, 1, n}], {n, 1, m}] + O[x]^m;
H[n_] := With[{g = 1 + x*G[n - 1, y]}, Sqrt[InverseSeries[x/g^2 + O[x]^(n + 1), x]/x]];
Join[{{1}, {0, 0}}, Append[CoefficientList[#, y], 0]& /@ CoefficientList[ H[11], x][[3;;]]] // Flatten (* Jean-François Alcover, Apr 15 2021, after Andrew Howroyd *)

\\ here G(n,y) gives A082680 as g.f.
G(n,y)={sum(n=1, n, x^n*sum(k=1, n, (n+k-1)!*(2*n-k)!*y^k/(k!*(n+1-k)!*(2*k-1)!*(2*n-2*k+1)!))) + O(x*x^n)}
H(n)={my(g=1+x*G(n-1, y), v=Vec(sqrt(serreverse(x/g^2)/x))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }

T(n,n+2-k) = T(n,k).

G.f.: A(x,y) satisfies A(x,y) = G(x*A(x,y)^2,y) where G(x,y) = 1 + x*B(x,y) and B(x,y) is the g.f. of A082680.

A342987 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 15, 5, 0, 4, 60, 84, 14, 0, 5, 175, 650, 420, 42, 0, 6, 420, 3324, 5352, 1980, 132, 0, 7, 882, 13020, 42469, 37681, 9009, 429, 0, 8, 1680, 42240, 246540, 429120, 239752, 40040, 1430, 0, 9, 2970, 118998, 1142622, 3462354, 3711027, 1421226, 175032, 4862

Offset: 0

Andrew Howroyd, Apr 03 2021

The number of vertices is n + 2 - k.

For k >= 2, column k is a polynomial of degree 4*(k-2)+1.

			Triangle begins:
  1;
  0, 1;
  0, 2,    2;
  0, 3,   15,     5;
  0, 4,   60,    84,     14;
  0, 5,  175,   650,    420,     42;
  0, 6,  420,  3324,   5352,   1980,    132;
  0, 7,  882, 13020,  42469,  37681,   9009,   429;
  0, 8, 1680, 42240, 246540, 429120, 239752, 40040, 1430;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, Table VIIIb.

Columns k=1..4 are A000007, A000027, A006470, A006471.
Diagonals are A000108, A002740, A006432, A006433.
Row sums are A342988.
Cf. A342981, A342982, A342984, A342985.

\\ here G(n,y) is A342984 as g.f.
F(n,y)={sum(n=0, n, x^n*sum(i=0, n, my(j=n-i); y^i*(2*i+2*j)!/(i!*(i+1)!*j!*(j+1)!))) + O(x*x^n)}
G(n,y)={my(g=F(n,y)); subst(g, x, serreverse(x*g^2))}
H(n)={my(g=G(n,y)-x, v=Vec(sqrt(serreverse(x/g^2)/x))); [Vecrev(t) | t<-v]}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }

G.f.: A(x,y) satisfies A(x,y) = G(x*A(x,y)^2,y) where G(x,y) + x is the g.f. of A342984.

A027836 Total number of vertices in all loopless rooted planar maps with n edges.

Original entry on oeis.org

1, 2, 8, 43, 268, 1824, 13156, 98865, 765948, 6075256, 49094708, 402801425, 3346590068, 28099903160, 238079915640, 2032914717645, 17476713955548, 151143219598008, 1314045772469632, 11478299163026540, 100688538612524720, 886622619082002120, 7834289222109530340

Offset: 0

Valery A. Liskovets

nonn

The number of rooted isthmusless n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005

L. M. Koganov, V. A. Liskovets, T. R. S. Walsh, Total vertex enumeration in rooted planar maps, Ars Combin. 54 (2000), 149-160.
V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Andrew Howroyd, Table of n, a(n) for n = 0..500
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Cf. A000260, A005470, A002293, A342981.

12*n*(4*n-1)!*(5*n^2+13*n+2)/(n!*(3*n+3)!);

Join[{1},Table[12n (4n-1)! (5n^2+13n+2)/(n!(3n+3)!),{n,20}]] (* Harvey P. Dale, May 20 2018 *)

a(n) = if(n==0, 1, 12*n*(4*n-1)!*(5*n^2+13*n+2)/(n!*(3*n+3)!)) \\ Andrew Howroyd, Apr 06 2021

a(n) = 12*n*(4*n-1)!*(5*n^2+13*n+2)/(n!*(3*n+3)!) for n > 0.
G.f.: -(1-3*g+g^2)*g where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011
a(n) = Sum_{k=1..n+1} k*A342981(n, k). - Andrew Howroyd, Apr 06 2021

Offset corrected and terms a(21) and beyond from Andrew Howroyd, Apr 06 2021

A006468 Number of rooted planar maps with 4 faces and n vertices and no isthmuses.

Original entry on oeis.org

5, 37, 150, 449, 1113, 2422, 4788, 8790, 15213, 25091, 39754, 60879, 90545, 131292, 186184, 258876, 353685, 475665, 630686, 825517, 1067913, 1366706, 1731900, 2174770, 2707965, 3345615, 4103442, 4998875, 6051169, 7281528, 8713232, 10371768, 12284965, 14483133, 16999206

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Simon Plouffe, Approximations of generating functions and a few conjectures, Master's thesis, UQAM, 1992; arXiv:0911.4975 [math.NT], 2009.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, Table VIb, f=5.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Column k=4 of A342981.

A006468[n_] := n*(n + 1)*(n + 2)*(n*(n*(2*n + 33) + 142) + 123)/360;
Array[A006468, 50] (* Paolo Xausa, Aug 20 2025 *)

a(n) = {n *(n+1) *(n+2) *(2*n^3 + 33*n^2 + 142*n + 123) /360} \\ Andrew Howroyd, Apr 02 2021

G.f.: -x*(x^3-4*x^2+2*x+5)/(x-1)^7, equivalent to a(n) = n *(n+1) *(n+2) *(2*n^3 +33*n^2 +142*n +123) /360, conjectured in Simon Plouffe's Master's thesis, 1992.

The above conjecture is true. - Andrew Howroyd, Apr 02 2021

Title improved and a(13)-a(14) from Sean A. Irvine, Apr 24 2017

Terms a(15) and beyond from Andrew Howroyd, Apr 02 2021

A006420 Number of rooted planar maps with 3 vertices and n faces and no isthmuses.

Original entry on oeis.org

1, 16, 150, 1104, 7077, 41504, 228810, 1205520, 6135690, 30391520, 147277676, 700990752, 3286733805, 15215673408, 69675615234, 316058238864, 1421891923038, 6350464644960, 28179908990772, 124327908683616, 545691921346146, 2383936774151616, 10370479696102500

Offset: 2

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 2..500
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259.

A diagonal of A342981.

seq(n)={my(g=x*(1-sqrt(1-4*x + O(x^n)))/(2*x)); Vec((1 + 2*g - 4*g^2)/((1 - g)^4*(1 - 2*g)^5))} \\ Andrew Howroyd, Apr 06 2021

G.f.: x^2*(1 + 2*g - 4*g^2)/((1 - g)^4*(1 - 2*g)^5) where g/x is the g.f. of A000108.

a(14) and a(15) from Sean A. Irvine, Apr 05 2017

Terms a(16) and beyond from Andrew Howroyd, Apr 02 2021

A006421 Number of rooted planar maps with 4 vertices and n faces and no isthmuses.

Original entry on oeis.org

1, 30, 449, 4795, 41850, 319320, 2213665, 14283280, 87169790, 508887860, 2865204762, 15654301865, 83388235348, 434685964540, 2223970137825, 11194499812388, 55546566721430, 272142754971892, 1318317357277470, 6321681903231990, 30037740651227756, 141545610360126400

Offset: 2

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 2..500
T. R. S. Walsh, A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259.

A diagonal of A342981.

seq(n)={my(g=x*(1-sqrt(1-4*x + O(x^n)))/(2*x)); Vec((1 + 9*g - 9*g^2 - 20*g^3 + 20*g^4)/((1 - g)^5*(1 - 2*g)^8))} \\ Andrew Howroyd, Apr 02 2021

G.f.: x^2*(1 + 9*g - 9*g^2 - 20*g^3 + 20*g^4)/((1 - g)^5*(1 - 2*g)^8) where g/x is the g.f. of A000108. - Andrew Howroyd, Apr 02 2021

a(13) and title improved by Sean A. Irvine, Apr 06 2017

Terms a(14) and beyond from Andrew Howroyd, Apr 02 2021

cp's OEIS Frontend

A000260 Number of rooted simplicial 3-polytopes with n+3 nodes; or rooted 3-connected triangulations with 2n+2 faces; or rooted 3-connected trivalent maps with 2n+2 vertices.

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A006419 a(n) = 2^(2*n+1) - C(2*n+3,n+1) + C(2*n+1,n).

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A343092 Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without isthmuses, n >= 2, k = 1..n-1.

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A342980 Triangle read by rows: T(n,k) is the number of rooted loopless planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

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A342987 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

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A027836 Total number of vertices in all loopless rooted planar maps with n edges.

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A006419 a(n) = 2^(2n+1) - C(2n+3,n+1) + C(2*n+1,n).