A290326 -id:A290326 - cp's OEIS frontend

A210252 Triangle read by rows: T(n,k) is the number of c-nets with n+1 faces and k+1 vertices, 1 <= k <= n. But see A290326 for a better version.

0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 3, 24, 0, 0, 0, 0, 33, 188, 0, 0, 0, 0, 13, 338, 1705, 0, 0, 0, 0, 0, 252, 3580, 16980, 0, 0, 0, 0, 0, 68, 3740, 39525, 180670, 0, 0, 0, 0, 0, 0, 1938, 51300, 452865, 2020120, 0, 0, 0, 0, 0, 0, 399, 38076, 685419, 5354832, 23478426, 0, 0, 0, 0, 0, 0, 0, 15180, 646415, 9095856, 65022840, 281481880, 0, 0, 0, 0, 0, 0, 0, 2530, 373175, 10215450, 120872850, 807560625, 3461873536, 0, 0, 0, 0, 0, 0, 0, 0, 121095, 7580040, 155282400, 1614234960, 10224817515, 43494961404

Offset: 1

N. J. A. Sloane, Mar 19 2012

c-nets are 3-connected rooted planar maps. This array also counts simple triangulations.
Table in Mullin & Schellenberg has incorrect values T(14,14) = 43494961412, T(15,13) = 21697730849, T(15,14) = 131631305614, T(15,15) = 556461655783. - Sean A. Irvine, Sep 28 2015
This triangle is based on a mis-reading of the Mullin-Schellenberg table. See A290326 for a better version. - N. J. A. Sloane, Jul 28 2017

			Triangle begins:
n\k
[1]  0
[2]  0 0
[3]  0 0 1
[4]  0 0 0 4
[5]  0 0 0 3 24
[6]  0 0 0 0 33 188
[7]  0 0 0 0 13 338 1705
[8]  0 0 0 0 0 252 3580 16980
[9]  0 0 0 0 0 68 3740 39525 180670
[10] 0 0 0 0 0 0 1938 51300 452865 2020120
[11] 0 0 0 0 0 0 399 38076 685419 5354832 23478426
[12] 0 0 0 0 0 0 0 15180 646415 9095856 65022840 281481880
[13] 0 0 0 0 0 0 0 2530 373175 10215450 120872850 807560625 3461873536
[14] 0 0 0 0 0 0 0 0 121095 7580040 155282400 1614234960 10224817515 43494961404
...

Gheorghe Coserea, Rows n = 1..100, flattened
R. C. Mullin, P. J. Schellenberg, The enumeration of c-nets via quadrangulations, J. Combinatorial Theory 4 1968 259--276. MR0218275 (36 #1362).

Right-hand edge is A001506.

See A290326 for a better version.

T(n,m) = {
  if (m <= 1+n\2 || n < 3, return(0));
  sum(k=0, m-1, sum(j=0, n-1,
     (-1)^((k+j+1)%2) * binomial(k+j,k)*(k+j+1)*(k+j+2)/2*
     (binomial(2*n, m-k-1) * binomial(2*m, n-j-1) -
      4 * binomial(2*n-1, m-k-2) * binomial(2*m-1, n-j-2))));
};
concat(vector(14, n, vector(n,m, T(n,m))))  \\ Gheorghe Coserea, Jan 08 2017

T(n,m) = Sum_{k=0..m-1} Sum_{j=0..n-1} (-1)^(k+j+1) * ((k+j+2)!/(2!*k!*j!)) * (binomial(2*n, m-k-1) * binomial(2*m, n-j-1) - 4 * binomial(2*n-1, m-k-2) * binomial(2*m-1, n-j-2)) if (n+2)/2 < m <= n and 0 otherwise. - Sean A. Irvine, Sep 28 2015

a(105)=T(14,14) corrected by Sean A. Irvine, Sep 28 2015

Name changed by Gheorghe Coserea, Jul 23 2017

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.

Original entry on oeis.org

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

A100960 Triangle read by rows: T(n,k) is the number of labeled 2-connected planar graphs with n nodes and k edges, n >= 3, n <= k <= 3(n-2).

Original entry on oeis.org

1, 3, 6, 1, 12, 70, 100, 45, 10, 60, 720, 2445, 3525, 2637, 1125, 195, 360, 7560, 46830, 132951, 210861, 205905, 123795, 40950, 5712, 2520, 84000, 835800, 3915240, 10549168, 18092368, 20545920, 15337560, 7193760, 1922760, 223440, 20160, 997920, 14757120, 103692960, 423918432, 1119730032, 2014030656, 2516883516, 2181661020, 1285377660, 491282820, 109907280, 10929600

Offset: 3

N. J. A. Sloane, Jan 12 2005

			The triangle T(n,k), n>=3, k>=3 begins:
  n\k [3] [4] [5] [6] [7]  [8]   [9]   [10]  [11]  [12]
  [3] 1;
  [4] 0,  3,  6,  1;
  [5] 0,  0,  12, 70, 100, 45,   10;
  [6] 0,  0,  0,  60, 720, 2445, 3525, 2637, 1125, 195;
  [7] ...

Gheorghe Coserea, Rows n=3..126, flattened
Edward A. Bender, Zhicheng Gao, and Nicholas C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.
Manuel Bodirsky, Clemens Gröpl, and Mihyun Kang, Generating Labeled Planar Graphs Uniformly At Random, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, Pages 377-386.
Jiaqi Liao, Hong Liu, and Guiying Yan, Isodiametric inequality for vector spaces, arXiv:2503.19239 [math.CO], 2025. See p. 8.

Cf. A267411, A290326.

Row sums give A096331. Main diagonal is A001710.

Q(n,k) = { \\ c-nets with n-edges, k-vertices
  if (k < 2+(n+2)\3 || k > 2*n\3, return(0));
  sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2*
  (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) -
  4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));
};
A100960_ser(N) = {
my(x='x+O('x^(3*N+1)), t='t+O('t^(N+4)),
   q=t*x*Ser(vector(3*N+1, n, Polrev(vector(min(N+3, 2*n\3), k, Q(n,k)),'t))),
   d=serreverse((1+x)/exp(q/(2*t^2*x) + t*x^2/(1+t*x))-1),
   g2=intformal(t^2/2*((1+d)/(1+x)-1)));
   serlaplace(Ser(vector(N, n, subst(polcoeff(g2, n,'t),'x,'t)))*'x);
};
A100960_seq(N) = {
  my(v=Vec(A100960_ser(N+2))); vector(#v, n, Vecrev(v[n]/t^(n+2)));
};
concat(A100960_seq(7)) \\ Gheorghe Coserea, Aug 09 2017

More terms from Michel Marcus, Feb 10 2016

A285165 Triangle read by rows: T(n,k) is the number of c-nets with n-k inner vertices and k outer vertices, 3 <= n, 2 <= k <= n-1.

Original entry on oeis.org

1, 1, 1, 7, 6, 1, 73, 56, 16, 1, 879, 640, 208, 30, 1, 11713, 8256, 2848, 560, 48, 1, 167423, 115456, 41216, 9440, 1240, 70, 1, 2519937, 1710592, 624384, 156592, 25864, 2408, 96, 1, 39458047, 26468352, 9812992, 2613664, 496944, 61712, 4256, 126, 1, 637446145, 423641088, 158883840, 44169600, 9234368, 1377600, 132480, 7008, 160, 1, 10561615871, 6966960128, 2636197888, 756712960, 169378560, 28663040, 3430528, 261648, 10920, 198, 1

Offset: 3

Gheorghe Coserea, Apr 12 2017

			Triangle starts:
n\k  [2]       [3]       [4]      [5]      [6]     [7]    [8]   [9]  [10]
[3]  1;
[4]  1,        1;
[5]  7,        6,        1;
[6]  73,       56,       16,      1;
[7]  879,      640,      208,     30,      1;
[8]  11713,    8256,     2848,    560,     48,     1
[9]  167423,   115456,   41216,   9440,    1240,   70,    1;
[10] 2519937,  1710592,  624384,  156592,  25864,  2408,  96,   1;
[11] 39458047, 26468352, 9812992, 2613664, 496944, 61712, 4256, 126, 1;
[12] ...

Gheorghe Coserea, Rows n=3..203, flattened
M. Bodirsky, C. Groepl, D. Johannsen and M. Kang, A Direct Decomposition of 3-connected Planar Graphs, conference paper (FPSAC05).

Cf. A290326.

Columns k=2-9 give: A106651(k=2), A285166(k=3), A285167(k=4), A285168(k=5), A285169(k=6), A285170(k=7), A285171(k=8), A285172(k=9).

x='x; y='y;
system("wget http://oeis.org/A106651/a106651.txt");
Fy = read("a106651.txt");
A106651_ser(N) = {
  my(y0 = 1 + O(x^N), y1=0, n=1);
  while(n++,
    y1 = y0 - subst(Fy, y, y0)/subst(deriv(Fy, y), y, y0);
    if (y1 == y0, break()); y0 = y1);
  y0;
};
z='z; t='t; u='u; c0='c0;
r1 = 2*t*u + 2*t^2*u + 2*t*u^2 + 2*t^2*u^2;
r2 = 4*t^2 + 4*t^3 + 4*t^2*u + 4*t^3*u;
r3 = -4*t^2 - 4*t^3 - 2*t*u - 6*t^2*u - 4*t^3*u - 2*t*u^2 - 2*t^2*u^2;
r4 = 2*t + 2*t^2 + 4*t^3 - u + t*u + 4*t^3*u + u^2 + t*u^2 - 2*t^2*u^2;
r5 = -2*t - 2*t^2 - 4*t^3 - 4*t*u - 2*t^2*u - 4*t^3*u + 2*t^2*u^2;
r6 = u + 2*t*u + 2*t^2*u - t*u^2;
Fz = r1*z^2 + (r3*c0 + r4)*z + r2*c0^2 + r5*c0 + r6;
seq(N) = {
  N += 10; my(z0 = 1 + O(t^N) + O(u^N), z1=0, n=1,
  Fz = subst(Fz, 'c0, subst(A106651_ser(N), 'x, 't)));
  while(n++,
    z1 = z0 - subst(Fz, z, z0)/subst(deriv(Fz, z) , z, z0);
    if (z1 == z0, break()); z0 = z1);
  vector(N-10, n, vector(n, k, polcoeff(polcoeff(z0, n-k), k-1)));
};
concat(seq(11))

A106651(n) = T(n,2) = Sum_{k=3..n-1} T(n,k), for n>=4.

T(n,n-2) = A054000(n-3) for n>= 5, T(n,n-3) = 8*A006325(n-3) for n>=6. - Gheorghe Coserea, Apr 19 2017

A099553 Number of rooted 2-connected loopless 4-regular planar maps with n inner faces.

Original entry on oeis.org

1, 2, 10, 42, 209, 1066, 5726, 31688, 180234, 1047356, 6198500, 37253790, 226891665, 1397880330, 8699804598, 54629525808, 345778883678, 2204263514460, 14142192816908, 91263177339092, 592069697914170, 3859674384409668, 25272938482712044

Offset: 3

N. J. A. Sloane, Nov 18 2004

nonn

a(n) is also the number of rooted loopless planar maps with n-1 edges and no isthmuses. - Andrew Howroyd, Apr 01 2021

a(n) is also the number of rooted 2-connected plane quadrangulations with n+1 vertices (allowing multiple edges). - Brendan McKay, Apr 08 2025

			A(x) = x^3 + 2*x^4 + 10*x^5 + 42*x^6 + 209*x^7 + 1066*x^8 + 5726*x^9 + ...

Gheorghe Coserea, Table of n, a(n) for n = 3..305
Han Ren, Yanpei Liu and Zhaoxiang Li, Enumeration of 2-connected Loopless 4-regular Maps on the Plane, European J. Combin., 23 (2002), 93-111.
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 (7).

Row sums of A342980.

Cf. A000139, A058860, A290326.

A099553 := proc(n)
    local e;
    e := n-1 ;
    add(binomial(2*e-r,e-2-2*r)*2^r*binomial(2*e,r),r=0..floor(e/2-1)) ;
    %-3*add(binomial(2*e-r,e-3-2*r)*2^r*binomial(2*e,r),r=0..floor((e-3)/2)) ;
    %*2/e ;
end proc:
seq(A099553(n),n=3..30) ; # R. J. Mathar, Aug 28 2018

a[n_] := Module[{e, s}, e = n-1; s = Sum[Binomial[2e-r, e-2-2r]*2^r*Binomial[2e, r], {r, 0, Floor[e/2-1]}]; s = s-3*Sum[Binomial[2e-r, e-3-2r]*2^r*Binomial[2e, r], {r, 0, Floor[(e-3)/2]}]; s=2s/e];
Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Feb 14 2023, after R. J. Mathar *)

F = (2*z^3*x^2 + (2*z^3 - 2*z)*x + (-z + 1))/(-2*z^3*x + 2*z);
G = x*(4*x + 1)*z^4 + 4*x*z^3 - 5*x*z^2 - 2*z + 2;
Z(N) = {
  my(z0=1+O('x^N), z1=0, n=1);
  while (n++,
    z1 = z0 - subst(G, 'z, z0)/subst(deriv(G,'z), 'z, z0);
    if (z1 == z0, break()); z0 = z1);
  z0;
};
seq(N) = Vec(subst(F, 'z, Z(N+3)));
seq(23)
\\ test: y = Ser(seq(303))*'x^3; 0 == 8*y^4 + (32*x + 12)*y^3 + (48*x^2 + 23*x + 6)*y^2 + (32*x^3 + 10*x^2 - 10*x + 1)*y + x^3*(8*x - 1)
\\ Gheorghe Coserea, Jul 13 2018

seq(n)={my(g=1+x*sum(n=1,n,x^n*binomial(3*n, n)*2/((n+1)*(2*n+1))) + O(x*x^n)); Vec(-1 + sqrt(serreverse(x/g^2)/x))} \\ Andrew Howroyd, Apr 06 2021

From Gheorghe Coserea, Jul 12 2018: (Start)
G.f. A(x) = (2*z^3*x^2 + (2*z^3 - 2*z)*x + (-z + 1))/(-2*z^3*x + 2*z), where z = 1 + 2*x^2 + 6*x^3 + 34*x^4 + 176*x^5 + 1004*x^6 + ... satisfies 0 = x*(4*x + 1)*z^4 + 4*x*z^3 - 5*x*z^2 - 2*z + 2. (See Theorem D in reference.)
G.f. y=A(x) satisfies:
0 = 8*y^4 + (32*x + 12)*y^3 + (48*x^2 + 23*x + 6)*y^2 + (32*x^3 + 10*x^2 - 10*x + 1)*y + x^3*(8*x - 1).
0 = x^3*(2*x + 1)*(49*x - 18)*(196*x - 27)*y'''' + x^2*(96040*x^3 - 27587*x^2 - 9297*x + 972)*y''' + (72030*x^4 - 36309*x^3 + 2010*x^2 - 864*x)*y'' - 6*(8*x + 3)*(49*x + 12)*y' + (2352*x + 576)*y.
(End)
Conjecture: 3*n *(3*n-1) *(5*n-8) *(3*n-2)*a(n) -(n-2) *(2*n-3) *(355*n^2 -703*n +300)*a(n-1) -98*(n-2) *(5*n-3) *(2*n-3) *(2*n-5) *a(n-2)=0. - R. J. Mathar, Aug 28 2018
G.f.: x*(A(x) - 1) where A(x) satisfies A(x) = G(x*A(x)^2) and (G(x) + 2*x - 1)/x is the g.f. of A000139. - Andrew Howroyd, Apr 06 2021

A291841 a(n) is the number of labeled 2-connected planar graphs with n edges.

Original entry on oeis.org

1, 3, 18, 131, 1180, 12570, 154525, 2150748, 33399546, 571979428, 10699844995, 216921707622, 4734437392728, 110613829184421, 2752971531611715, 72676980383698345, 2027560176161932735, 59579981648921326791, 1838669555339295257097, 59435431024069408426431

Offset: 3

Gheorghe Coserea, Sep 10 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 3..123
E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.

Cf. A096331, A290326.

Column sums of A100960.

Q(n,k) = { \\ c-nets with n-edges, k-vertices
  if (k < 2+(n+2)\3 || k > 2*n\3, return(0));
  sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2*
  (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) -
  4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));
};
seq(N) = {
my(x='x+O('x^(N+3)), t='t,
   q=t*x*Ser(vector(N, n, Polrev(vector(2*n\3, k, Q(n,k)),t))),
   d=serreverse((1+x)/exp(q/(2*t^2*x) + t*x^2/(1+t*x))-1),
   g2=intformal(t^2/2*((1+d)/(1+x)-1)), e2=apply(serlaplace, g2));
   Vec(subst(e2, 't, 1));
};
seq(22)

A001506 a(n) is the number of c-nets with n+1 vertices and 2n edges, n >= 1.

Original entry on oeis.org

0, 0, 1, 4, 24, 188, 1705, 16980, 180670, 2020120, 23478426, 281481880, 3461873536, 43494961404, 556461656569, 7230987646484, 95244774132810, 1269534571172912, 17100621281619328, 232511930087682528, 3188042426888493288

Offset: 1

N. J. A. Sloane

nonn

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).

Gheorghe Coserea, Table of n, a(n) for n = 1..200
R. C. Mullin and P. J. Schellenberg, The enumeration of c-nets via triangulations, J. Combin. Theory, 4 (1968), 259-276.

Cf. A290326.

A290326(n,k) = {
  if (n < 3 || k < 3, return(0));
  sum(i=0, k-1, sum(j=0, n-1,
     (-1)^((i+j+1)%2) * binomial(i+j, i)*(i+j+1)*(i+j+2)/2*
     (binomial(2*n, k-i-1) * binomial(2*k, n-j-1) -
      4 * binomial(2*n-1, k-i-2) * binomial(2*k-1, n-j-2))));
};
vector(21, n, A290326(n,n))  \\ Gheorghe Coserea, Jul 28 2017

a(n) = A290326(n,n). - Sean A. Irvine, Sep 29 2015

Corrected and extended by Sean A. Irvine, Sep 29 2015

Name changed by Gheorghe Coserea, Jul 23 2017

A288265 Triangle read by rows: T(n,k) is the number of labeled connected planar graphs on n vertices and k edges.

Original entry on oeis.org

1, 1, 3, 1, 16, 15, 6, 1, 125, 222, 205, 120, 45, 10, 1296, 3660, 5700, 6165, 4935, 2937, 1125, 195, 16807, 68295, 156555, 258125, 330456, 334530, 254275, 131985, 40950, 5712, 262144, 1436568, 4483360, 10230360, 18528216, 27261192, 31761744, 27958920, 17666320, 7513632, 1922760, 223440, 4782969, 33779340, 136368414, 405918324, 970196283, 1910996136, 3058785990, 3866563764, 3754432899, 2724326136, 1425385584, 507370500, 109907280, 10929600

Offset: 1

Gheorghe Coserea, Aug 14 2017

Row n >= 3 contains 3*n-5 terms.

			A(x;t) = x + t*x^2/2! + (3*t^2 + t^3)*x^3/3! + (16*t^3 + 15*t^4 + 6*t^5 + t^6)*x^4/4! + ...
Triangle starts:
n\k [0] [1] [2] [3] [4]  [5]   [6]   [7]   [8]   [9]   [10]  [11]  [12]
[1] 1;
[2] 0   1;
[3] 0,  0,  3,  1;
[4] 0,  0,  0,  16, 15,  6,    1;
[5] 0,  0,  0,  0,  125, 222,  205,  120,  45,   10;
[6] 0,  0,  0,  0,  0,   1296, 3660, 5700, 6165, 4935, 2937, 1125, 195;
[7] ...

Gheorghe Coserea, Rows n = 1..126, flattened
E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, Pages 377-386.
Omer Gimenez, Marc Noy, Asymptotic enumeration and limit laws of planar graphs, J. Amer. Math. Soc. 22 (2009), 309-329.

Cf. A000272, A007816, A096332, A100960, A266390, A267411, A288266, A290326.

Q(n,k) = { \\ c-nets with n-edges, k-vertices
  if (k < 2+(n+2)\3 || k > 2*n\3, return(0));
  sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2*
  (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) -
  4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));
};
A100960_ser(N) = {
my(x='x+O('x^(3*N+1)), t='t+O('t^(N+4)),
   q=t*x*Ser(vector(3*N+1, n, Polrev(vector(min(N+3, 2*n\3), k, Q(n,k)),'t))),
   d=serreverse((1+x)/exp(q/(2*t^2*x) + t*x^2/(1+t*x))-1),
   g2=intformal(t^2/2*((1+d)/(1+x)-1)));
   serlaplace(Ser(vector(N, n, subst(polcoeff(g2, n,'t),'x,'t)))*'x);
};
A288265_ser(N) = {
  my(x='x+O('x^(N+3)), b = t*x^2/2 + serconvol(A100960_ser(N), exp(x)),
     g1=intformal(serreverse(x/exp(b'))/x)); serlaplace(g1);
};
A288265_seq(N) = {
  my(v=Vec(A288265_ser(N))); vector(#v, n, Vecrev(v[n]/t^(n-1)));
};
concat(A288265_seq(9))

A096332(n) = Sum_{k=n-1..3*n-6} T(n,k) for n >= 3.

A000272(n) = T(n,n-1), A007816(n-3) = T(n, 3*n-6).

cp's OEIS Frontend

A210252 Triangle read by rows: T(n,k) is the number of c-nets with n+1 faces and k+1 vertices, 1 <= k <= n. But see A290326 for a better version.

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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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A100960 Triangle read by rows: T(n,k) is the number of labeled 2-connected planar graphs with n nodes and k edges, n >= 3, n <= k <= 3(n-2).

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A285165 Triangle read by rows: T(n,k) is the number of c-nets with n-k inner vertices and k outer vertices, 3 <= n, 2 <= k <= n-1.

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A099553 Number of rooted 2-connected loopless 4-regular planar maps with n inner faces.

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A291841 a(n) is the number of labeled 2-connected planar graphs with n edges.

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A001506 a(n) is the number of c-nets with n+1 vertices and 2n edges, n >= 1.

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