A000260 -id:A000260 - cp's OEIS frontend

A000256 Number of simple triangulations of the plane with n nodes.

1, 1, 0, 1, 3, 12, 52, 241, 1173, 5929, 30880, 164796, 897380, 4970296, 27930828, 158935761, 914325657, 5310702819, 31110146416, 183634501753, 1091371140915, 6526333259312, 39246152584304, 237214507388796, 1440503185260748

Offset: 3

N. J. A. Sloane

A triangulation is simple if it contains no separating 3-cycle. The triangulations are rooted with three fixed exterior nodes. - Andrew Howroyd, Feb 24 2021

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, pp. 437-448 of J. N. Srivastava, ed., A Survey of Combinatorial Theory, North-Holland, 1973.

T. D. Noe, Table of n, a(n) for n=3..200
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.
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. FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics; arXiv:0912.0072 [math.NT], 2009.
P. N. Rathie, A census of simple planar triangulations, J. Combin. Theory, B 16 (1974), 134-138.
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.
William T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

First row of array in A210664.

R := RootOf(x-t*(t-1)^2, t); ogf := series( (2*R^3+2*R^2-2*R-1)/((R-1)*(R+1)^2), x=0, 20); # Mark van Hoeij, Nov 08 2011

r = Root[x - t*(t - 1)^2, t, 1] ; CoefficientList[ Series[(2*r^3 + 2*r^2 - 2*r - 1)/((r - 1)*(r + 1)^2), {x, 0, 24}], x] (* Jean-François Alcover, Mar 14 2012, after Maple *)

A000260_ser(N) = {
  my(v = vector(N, n, binomial(4*n+2, n+1)/((2*n+1)*(3*n+2))));
  Ser(concat(1,v));
};
A000256_seq(N) = {
  my(g = A000260_ser(N)); Vec(subst(2 - 1/g, 'x, serreverse(x*g^2)));
};
A000256_seq(24)
\\ test: y = Ser(A000256_seq(200)); 0 == x*(x+4)^2*y^3 - x*(6*x^2+51*x+76)*y^2 + (12*x^3+108*x^2+115*x-1)*y - (8*x^3+76*x^2+54*x-1)
\\ Gheorghe Coserea, Jul 31 2017

seq(n)={my(g=1+serreverse(x/(1+x)^4 + O(x*x^n) )); Vec(2 - sqrt(serreverse( x*(2-g)^2*g^4)/x ))} \\ Andrew Howroyd, Feb 23 2021

a(n) = (1/4)*(7*binomial(3*n-9, n-4)-(8*n^2-43*n+57)*a(n-1)) / (8*n^2-51*n+81), n>4. - Vladeta Jovovic, Aug 19 2004
(1/4 + 7/8*n - 9/8*n^3)*a(n) + (-5/4 + 2/3*n + 59/12*n^2 - 13/3*n^3)*a(n+1) + (-1 - 2/3*n + n^2 + 2/3*n^3)*a(n+2). - Simon Plouffe, Feb 09 2012
a(n) ~ 3^(3*n-6+1/2)/(2^(2*n+3)*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Aug 13 2013
From Gheorghe Coserea, Jul 31 2017: (Start)
G.f. y(x) satisfies (with offset 0):
y(x*g^2) = 2 - 1/g, where g=A000260(x). (eqn 2.6 in Tutte's paper)
0 = x*(x+4)^2*y^3 - x*(6*x^2+51*x+76)*y^2 + (12*x^3+108*x^2+115*x-1)*y - (8*x^3+76*x^2+54*x-1).
0 = x*(27*x-4)*deriv(y,x) + x*(7*x+28)*y^2 - 2*(14*x^2+45*x+1)*y + 2*(14*x^2+34*x+1).
(End)

More terms from Vladeta Jovovic, Aug 19 2004

A006390 Number of sensed loopless planar maps with n edges.

Original entry on oeis.org

1, 1, 2, 5, 14, 49, 240, 1259, 7570, 47996, 319518, 2199295, 15571610, 112773478, 832809504, 6253673323, 47650870538, 367784975116, 2871331929096, 22647192990256, 180277915464664, 1447060793168493, 11703567787559680, 95312765368320637, 781151020141584190

Offset: 0

N. J. A. Sloane

nonn

By duality, also the number of sensed isthmusless planar maps with n edges. An isthmus may also be called a bridge. - Andrew Howroyd, Mar 28 2021

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 = 0..500
V. A. Liskovets and T. R. S. Walsh, Counting Unrooted Loopless Planar Maps [Extended abstract]
V. A. Liskovets and T. R. S. Walsh, Counting unrooted loopless planar maps, Europ. J. Combin., 26:5 (2005), 651-663.
Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.

Cf. A000010, A006384, A000260 (rooted), A006391 (unsensed case), A103941 (with distinguished face), A103942 (with distinguished vertex).

a[n_] := If[n==0, 1, (1/(2n))(Sum[Binomial[4k, k] EulerPhi[n/k] Boole[ 0Jean-François Alcover, Aug 29 2019 *)

a(n) = {if(n==0, 1, (sumdiv(n, d, if(dAndrew Howroyd, Mar 28 2021

a(n) = (1/(2n))*[2(4n+1)*binomial(4n, n)/((n+1)*(3n+1)*(3n+2)) + Sum_{0A000010), q(n)=binomial(2n, (n-2)/2) if n is even and q(n)=2n*binomial(2n, (n-1)/2)/(n+1) if n is odd.

More terms from Valery A. Liskovets, Dec 01 2003

a(17) and a(19) corrected by Sean A. Irvine, Mar 26 2017

A197271 a(n) = (10 / ((3n+1)(3n+2))) binomial(4*n, n).

Original entry on oeis.org

5, 2, 5, 20, 100, 570, 3542, 23400, 161820, 1159400, 8544965, 64448228, 495508780, 3872033900, 30680401500, 246041115600, 1993987498284, 16310419381080, 134519771966180, 1117653277802000, 9347742311507600, 78652006531467930, 665393840873409150, 5657273782416664200, 48318619683648190500

Offset: 0

Peter Bala, Oct 12 2011

A combinatorial interpretation for this sequence in terms of a family of plane trees is given in [Schaeffer, Corollary 2 with k = 4].

For n>=1, the number of rooted strict triangulations of a square with n-1 internal vertices, where a triangulation is "strict" if no two distinct edges have the same pair of ends. See equation (1) in [Tutte 1980] (who references [Brown 1964]) for the number of rooted strict near-triangulations of type (n,m), with m=1. - Noam Zeilberger, Jan 04 2023

Michael De Vlieger, Table of n, a(n) for n = 0..1031
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
W. G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]
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.
G. Schaeffer, A combinatorial interpretation of super-Catalan numbers of order two, (2001).
William T. Tutte, On the enumeration of convex polyhedra, J. Combin. Theory Ser. B 28 (1980), 105-126.

Column m=1 of A146305.

Cf. A000139, A000260, A197272.

Table[10/((3n+1)(3n+2)) Binomial[4n,n],{n,0,30}] (* Harvey P. Dale, Jan 27 2015 *)

a(n) = 10/((3*n+1)*(3*n+2))*binomial(4*n,n).
a(n) = A000260(n) * 5*(n+1)/(4*n+1). - Noam Zeilberger, May 20 2019
a(n) ~ c*(256/27)^n / n^(5/2), where c = (10/9)*sqrt(2/(3*Pi)) = 0.511843.... - Peter Luschny, Jan 05 2023
D-finite with recurrence 3*n*(3*n+2)*(3*n+1)*a(n) -8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Jul 31 2024

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

A002712 Number of unrooted triangulations of a disk that have reflection symmetry with n interior nodes and 3 nodes on the boundary.

Original entry on oeis.org

1, 1, 1, 3, 8, 23, 68, 215, 680, 2226, 7327, 24607, 83060, 284046, 975950, 3383343, 11778308, 41269252, 145131502, 512881550, 1818259952, 6470758289, 23091680690, 82659905947, 296605398856, 1067012168350, 3846553544904, 13896522968160, 50296815014780, 182378110257354, 662384549806938

Offset: 0

N. J. A. Sloane

nonn

These are also called [n,0]-triangulations.

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).

Andrew Howroyd, Table of n, a(n) for n = 0..500
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
W. G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]
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)

Column k=0 of A169809.

Dc := proc(n,m) 2*(2*m+3)!*(4*n+2*m+1)!/m!/(m+2)!/n!/(3*n+2*m+3)! ; end:
A000260 := proc(n) Dc(n,0) ; end:
Dx2 := proc(nmax) add( A000260(n)*x^(2*n),n=0..nmax) ; end:
o := 20: Order := 2*o-1 : j := solve( J0=1+x*J0+x^2*J0*(1+x*J0/2)*series(J0^2-Dx2(o),x=0,2*o-1),J0) ;
for n from 0 to 2*o-2 do printf("%d,",coeftayl(j,x=0,n)) ; od: # R. J. Mathar, Oct 29 2008

seq[m_] := Module[{q}, q = Sum[x^(2n) Binomial[4n+2, n+1]/ ((2n+1)(3n+2)), {n, 0, Quotient[m, 2]}]; p = 1+O[x]; Do[p = 1 + x*p + x^2*p*(1+x*p/2)(p^2-q), {n, 1, m}]; CoefficientList[p, x]];
seq[30] (* Jean-François Alcover, Apr 25 2023, after Andrew Howroyd *)

seq(n)={my(q=sum(n=0, n\2, x^(2*n)*binomial(4*n+2, n+1)/((2*n+1)*(3*n+2))), p=1+O(x)); for(n=1, n, p = 1 + x*p + x^2*p*(1 + x*p/2)*(p^2 - q)); Vec(p)} \\ Andrew Howroyd, Feb 24 2021

More terms from R. J. Mathar, Oct 29 2008

Name clarified and terms a(27) and beyond from Andrew Howroyd, Feb 24 2021

A007767 Number of pairs of permutations of degree n that avoid (12,21).

Original entry on oeis.org

1, 1, 3, 17, 151, 1899, 31711, 672697, 17551323, 549500451, 20246665349, 864261579999, 42190730051687, 2329965898878307, 144220683681814515, 9926440976428215117, 754465679498026783923, 62939664181821196179459, 5732069150321309755351161, 567176164248814234096702451

Offset: 0

Steve Linton

nonn

A pair of permutations (p,q) of degree n avoid (12,21) if there do not exist indices 1<=iNoam Zeilberger, Jun 06 2016 (via Steve Linton)

Number of intervals (i.e. ordered pairs (x,y) such that x<=y) in the permutation lattice of size n, that is, pairs of permutations (x,y) related by the weak Bruhat order x<=y iff inversions(x) is a subset of inversions(y) (see Hammett and Pittel, p. 4567). - Noam Zeilberger, Jun 01 2016

Andrew Elvey Price, Table of n, a(n) for n = 0..26
Noga Alon, Kirill Rudov, and Leeat Yariv, Dominance Solvability in Random Games, Princeton Univ. (2020).
Noga Alon, Kirill Rudov, and Leeat Yariv, Online Appendix for 'Dominance Solvability in Random Games', Princeton Univ. (2021).
Grégory Chatel, Vincent Pilaud, and Viviane Pons, The weak order on integer posets, arXiv:1701.07995 [math.CO], 2017.
Clément Chenevière, Enumerative study of intervals in lattices of Tamari type, Ph. D. thesis, Univ. Strasbourg (France), Ruhr-Univ. Bochum (Germany), HAL tel-04255439 [math.CO], 2024. See pp. 3, 33, 145.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
Benjamin Gunby, Asymptotics of Pattern Avoidance in the Permutation-Tuple and Klazar Set Partition Settings, arXiv:1609.06023 [math.CO], 2017.
Adam Hammett and Boris Pittel, How often are two permutations comparable?, Transactions of the AMS 360:9 (2008), 4541-4568.
Evgeny Kapun, Java program for generating terms a(0)-a(13).

Cf. A000260, A263754.

Java
```
// See Kapun link.
```

a(n) = Sum_{k=1..n!} k * A263754(n,k). - Alois P. Heinz, Jun 06 2016

a(0)=1 prepended by Alois P. Heinz, Jun 06 2016
a(10)-a(13) from Evgeny Kapun, Dec 11 2016
More terms from Andrew Elvey Price, Feb 08 2024

A006391 Number of unsensed loopless planar maps with n edges.

Original entry on oeis.org

1, 1, 2, 5, 14, 45, 191, 871, 4682, 27336, 172706, 1150322, 7989004

Offset: 0

N. J. A. Sloane

nonn

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

Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.

Cf. A006385, A000260 (rooted), A006390 (sensed).

a(8)-a(12) from Sean A. Irvine, Mar 29 2017

a(0)=1 prepended by Andrew Howroyd, Jan 14 2025

A058859 Number of 1-connected rooted cubic planar maps with n faces.

Original entry on oeis.org

1, 3, 19, 143, 1089, 8564, 69075, 569469, 4783377, 40829748, 353395155, 3096104105, 27415923905, 245069538465, 2209155012387, 20064713628389, 183478258249569, 1688112897834496, 15618577076864579, 145242456429736935

Offset: 4

N. J. A. Sloane, Jan 06 2001; revised Feb 17 2006

nonn

Gheorghe Coserea, Table of n, a(n) for n = 4..306
Z. Gao and N. C. Wormald, Enumeration of rooted cubic planar maps
Z. Gao and N. C. Wormald, Enumeration of rooted cubic planar maps, Annals of Combinatorics, 6 (2002), no. 3-4, 313-325.

Cf. A000260, A058860, A058861.

eq:=16*x^11*m^4+(-24*x^9+32*x^8+72*x^7)*m^3+(-15*x^7-108*x^6-194*x^5-92*x^4+x^3)*m^2+(-2*x^5-33*x^4-70*x^3-46*x^2+16*x-1)*m-x^2-11*x+1: m:=sum(A[j]*x^j,j=0..35): A[0]:=solve(subs(x=0,expand(eq))): for n from 1 to 35 do A[n]:=solve(coeff(expand(eq),x^n)=0) od: C:=(1-2*x-4*x^2)*x^4*m-2*x^8*m^2: Cser:=series(C,x=0,30): seq(coeff(Cser,x^n),n=4..26); # Emeric Deutsch, Nov 30 2005

F = x^4*(1-2*x-4*x^2)*z - 2*x^8*z^2;
G = 16*x^11*z^4 - 8*x^7*(3*x^2 - 4*x - 9)*z^3 - x^3*(15*x^4 + 108*x^3 + 194*x^2 + 92*x - 1)*z^2 -  (2*x^5 + 33*x^4 + 70*x^3 + 46*x^2 - 16*x + 1)*z - x^2 - 11*x + 1;
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)));
seq(20)
\\ test: y = Ser(seq(303))*'x^4; 0 == 64*y^4 + (912*x^4 + 640*x^3 + 384*x^2 + 3328*x + 2864)*y^3 - (1743*x^8 + 13968*x^7 + 13344*x^6 - 52888*x^5 - 116934*x^4 - 71248*x^3 - 4064*x^2 + 3768*x - 41)*y^2 + (784*x^12 + 13524*x^11 + 29478*x^10 - 51033*x^9 - 194686*x^8 - 166400*x^7 - 5454*x^6 + 43746*x^5 + 4030*x^4 - 5652*x^3 + 904*x^2 - 41*x)*y - x^5*(x^2 + 11*x - 1)*(1568*x^8 + 476*x^7 - 7456*x^6 - 8458*x^5 - 27*x^4 + 2672*x^3 + 130*x^2 - 330*x + 41)
\\ Gheorghe Coserea, Jul 15 2018

G.f.: x^4*(1-2*x-4*x^2)*m-2*x^8*m^2, where m is defined by 16*x^11*m^4 + (-24*x^9+32*x^8+72*x^7)*m^3 + (-15*x^7-108*x^6-194*x^5-92*x^4+x^3)*m^2 + (-2*x^5-33*x^4-70*x^3-46*x^2+16*x-1)*m - x^2-11*x+1=0. - Emeric Deutsch, Nov 30 2005
From Gheorghe Coserea, Jul 16 2018: (Start):
G.f. y=A(x) satisfies:
0 = 64*y^4 + (912*x^4 + 640*x^3 + 384*x^2 + 3328*x + 2864)*y^3 - (1743*x^8 + 13968*x^7 + 13344*x^6 - 52888*x^5 - 116934*x^4 - 71248*x^3 - 4064*x^2 + 3768*x - 41)*y^2 + (784*x^12 + 13524*x^11 + 29478*x^10 - 51033*x^9 - 194686*x^8 - 166400*x^7 - 5454*x^6 + 43746*x^5 + 4030*x^4 - 5652*x^3 + 904*x^2 - 41*x)*y - x^5*(x^2 + 11*x - 1)*(1568*x^8 + 476*x^7 - 7456*x^6 - 8458*x^5 - 27*x^4 + 2672*x^3 + 130*x^2 - 330*x + 41).
0 = x*(4*x^2 + 8*x + 5)*(27*x^6 + 216*x^5 + 171*x^4 - 208*x^3 - 339*x^2 + 24*x + 1)*(53687232*x^17 + 962429472*x^16 + 4910442696*x^15 + 11262716564*x^14 + 13535708340*x^13 + 6699339314*x^12 - 8161216832*x^11 - 27707772057*x^10 - 38282906893*x^9 - 23841839272*x^8 + 3164178022*x^7 + 13551725887*x^6 + 6618789645*x^5 + 110368160*x^4 - 189595230*x^3 + 52114000*x^2 - 2282040*x - 80000)*y'''' - (23192884224*x^25 + 642325749120*x^24 + 7010404371072*x^23 + 38396140051536*x^22 + 119087871158520*x^21 + 209055666121344*x^20 + 149537518315396*x^19 - 179206877652920*x^18 - 594068689834972*x^17 - 713069283397760*x^16 - 388115755832091*x^15 + 185412410945637*x^14 + 709124462066474*x^13 + 898548947063912*x^12 + 629038710881040*x^11 + 159866881148998*x^10 - 107640739893374*x^9 - 101244290972424*x^8 - 23418947186993*x^7 + 3644481830365*x^6 + 957436398080*x^5 - 94641974160*x^4 + 1607421440*x^3 + 430075760*x^2 - 17060400*x - 400000)*y''' + (69578652672*x^24 + 1910859372288*x^23 + 21034975582656*x^22 + 114742977687936*x^21 + 350375920009560*x^20 + 585065268522672*x^19 + 317856584972580*x^18 - 736872920930424*x^17 - 1812132349221252*x^16 - 1696870248263700*x^15 - 376785528937023*x^14 + 1026609868750112*x^13 + 1799851001684942*x^12 + 1902275760186412*x^11 + 1364464778889680*x^10 + 504031822062384*x^9 - 75374914747162*x^8 - 173636873122824*x^7 - 67965626046313*x^6 - 3235617436480*x^5 + 1670710238920*x^4 - 60241392600*x^3 - 9066655340*x^2 + 1117875760*x + 15179600)*y'' - 12*(11596442112*x^23 + 315790249536*x^22 + 3414867276384*x^21 + 17899179378120*x^20 + 51714502467480*x^19 + 77928289056012*x^18 + 22675972179932*x^17 - 134244171463804*x^16 - 254323096657040*x^15 - 181481980531415*x^14 + 24427607774667*x^13 + 176309477492908*x^12 + 214672437288248*x^11 + 192416432064275*x^10 + 135698454441595*x^9 + 59484339948854*x^8 + 1838501691038*x^7 - 16090673029130*x^6 - 8704257466200*x^5 - 1085436408240*x^4 + 33590844600*x^3 - 6624333760*x^2 - 719889600*x - 8800000)*y' + 12*(11596442112*x^22 + 313103937024*x^21 + 3232316223360*x^20 + 15530584062240*x^19 + 39522162905640*x^18 + 45540724655832*x^17 - 16695945361396*x^16 - 123726467878420*x^15 - 152050336659260*x^14 - 49261893247550*x^13 + 73707236060447*x^12 + 119787972312984*x^11 + 115583117491500*x^10 + 95686381642950*x^9 + 56811985465335*x^8 + 13932882885644*x^7 - 9032398496482*x^6 - 8810946218840*x^5 - 1354608403560*x^4 + 47155824160*x^3 - 6777547760*x^2 - 855133760*x - 10609600)*y.
(End)

More terms from Emeric Deutsch, Nov 30 2005

A058861 Number of 3-connected rooted cubic planar maps with n faces and girth at least 4.

Original entry on oeis.org

0, 0, 1, 3, 12, 59, 313, 1713, 9559, 54189, 311460, 1812281, 10661303, 63336873, 379601353, 2293205687, 13953099573, 85451824382, 526431271347, 3260689089300, 20296848348929, 126918850161182, 796981464813540

Offset: 4

N. J. A. Sloane, Jan 06 2001; revised Feb 17 2006

nonn

Number of 3-connected triangle-free rooted cubic maps with n faces.

Gheorghe Coserea, Table of n, a(n) for n = 4..308
Z. Gao and N. C. Wormald, Enumeration of rooted cubic planar maps
Z. Gao and N. C. Wormald, Enumeration of rooted cubic planar maps, Annals of Combinatorics, 6 (2002), no. 3-4, 313-325.

Cf. A000260, A058859, A058860.

eq:=(x^3-3*x^2+3*x-1)*g^4+(4*x^4-12*x^3+9*x^2+2*x-3)*g^3+(6*x^5-10*x^4-15*x^3+36*x^2-14*x-3)*g^2+(4*x^6+4*x^5-45*x^4+82*x^3-59*x^2+14*x-1)*g+x^7+5*x^6-8*x^5+x^4: g:=sum(A[j]*x^j,j=1..37): for n from 1 to 37 do A[n]:=solve(coeff(expand(eq),x^n)=0) od: C3:=x^2*(1-3*x)*g: C3ser:=series(C3,x=0,34): seq(coeff(C3ser,x^n),n=6..30); # Emeric Deutsch, Nov 30 2005

F = x^2*(1 - 3*x)*z;
G = x^12*(x - 1)^3*z^4 + x^8*(x - 1)^2*(2*x - 3)*(2*x + 1)*z^3 + x^4*(x - 1)*(6*x^4 - 4*x^3 - 19*x^2 + 17*x + 3)*z^2 + (4*x^6 + 4*x^5 - 45*x^4 + 82*x^3 - 59*x^2 + 14*x - 1)*z + (x^3 + 5*x^2 - 8*x + 1);
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) = concat([0, 0], Vec(subst(F, 'z, 'x^4*Z(N))));
seq(21)
\\ test: y = Ser(seq(303))*'x^4; 0 == (x - 1)^3*y^4 - x^2*(x - 1)^2*(2*x - 3)*(2*x + 1)*(3*x - 1)*y^3 + x^4*(x - 1)*(3*x - 1)^2*(6*x^4 - 4*x^3 - 19*x^2 + 17*x + 3)*y^2 - x^6*(3*x - 1)^3*(4*x^6 + 4*x^5 - 45*x^4 + 82*x^3 - 59*x^2 + 14*x - 1)*y + x^12*(3*x - 1)^4*(x^3 + 5*x^2 - 8*x + 1)
\\ Gheorghe Coserea, Jul 15 2018

G.f.: x^2*(1-3*x)*g, where g is defined by (x^3-3*x^2+3*x-1)*g^4 + (4*x^4-12*x^3+9*x^2+2*x-3)*g^3 + (6*x^5-10*x^4-15*x^3+36*x^2-14*x-3)*g^2 + (4*x^6+4*x^5-45*x^4+82*x^3-59*x^2+14*x-1)*g + x^7+5*x^6-8*x^5+x^4=0. - Emeric Deutsch, Nov 30 2005
From Gheorghe Coserea, Jul 15 2018: (Start)
G.f. y=A(x) satisfies:
0 = (x - 1)^3*y^4 - x^2*(x - 1)^2*(2*x - 3)*(2*x + 1)*(3*x - 1)*y^3 + x^4*(x - 1)*(3*x - 1)^2*(6*x^4 - 4*x^3 - 19*x^2 + 17*x + 3)*y^2 - x^6*(3*x - 1)^3*(4*x^6 + 4*x^5 - 45*x^4 + 82*x^3 - 59*x^2 + 14*x - 1)*y + x^12*(3*x - 1)^4*(x^3 + 5*x^2 - 8*x + 1).
0 = x^4*(x - 1)^3*(3*x - 1)^4*(256*x^3 - 512*x^2 + 256*x - 27)*(660*x^5 - 2668*x^4 + 4177*x^3 - 3252*x^2 + 1305*x - 220)*y'''' - 4*x^3*(x - 1)^2*(3*x - 1)^3*(696960*x^10 - 5684352*x^9 + 19870624*x^8 - 39218578*x^7 + 48478923*x^6 - 39311914*x^5 + 21210466*x^4 - 7501069*x^3 + 1650104*x^2 - 201370*x + 10395)*y''' + 4*x^2*(x - 1)*(3*x - 1)^2*(7579440*x^12 - 77500656*x^11 + 341548428*x^10 - 862746936*x^9 + 1396393806*x^8 - 1530275829*x^7 + 1167408906*x^6 - 625929723*x^5 + 234247228*x^4 - 59616890*x^3 + 9784582*x^2 - 931830*x + 38940)*y'' - 24*x*(3*x - 1)*(8523900*x^14 - 105292620*x^13 + 561229815*x^12 - 1731677190*x^11 + 3479254732*x^10 - 4837165728*x^9 + 4815815835*x^8 - 3498631418*x^7 + 1868972298*x^6 - 732379803*x^5 + 207693098*x^4 - 41430916*x^3 + 5510394*x^2 - 438095*x + 15730)*y' + 24*(27442800*x^15 - 372895380*x^14 + 2140330050*x^13 - 7047776880*x^12 + 15074631336*x^11 - 22357962673*x^10 + 23891962029*x^9 - 18825921582*x^8 + 11080006886*x^7 - 4892373579*x^6 + 1614037497*x^5 - 392156906*x^4 + 68130318*x^3 - 8004294*x^2 + 569210*x - 18480)*y.
(End)

More terms from Emeric Deutsch, Nov 30 2005

A103941 Number of unrooted loopless n-edge maps in the plane (planar with a distinguished outside face).

Original entry on oeis.org

1, 1, 2, 6, 22, 103, 614, 3872, 26414, 186988, 1367976, 10254326, 78461338, 610598818, 4821248244, 38546510368, 311560875422, 2542507084588, 20925300483992, 173530381632724, 1448900079476152, 12172334379246523, 102833593763830038, 873187910184763024, 7449120536014301138

Offset: 0

Valery A. Liskovets, Mar 17 2005

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. A002293, A006390, A103942, A000260, A005470.

a[n_] := (1/(2n)) (Binomial[4n, n]/(3n+1) + Sum[Boole[0 < k < n] EulerPhi[ n/k] Binomial[4k, k], {k, Divisors[n]}] + q[n]);
q[n_] := If[EvenQ[n], 0, Binomial[2n, (n-1)/2]];
Array[a, 20] (* Jean-François Alcover, Sep 01 2019 *)

a(n) = {if(n==0, 1, (sumdiv(n, d, if(dAndrew Howroyd, Mar 28 2021

For n > 0, a(n) = (1/(2n))*[binomial(4n, n)/(3n+1) + Sum_{0A000010, q(n)=0 if n is even and q(n)=binomial(2n, (n-1)/2) if n is odd.

a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Mar 28 2021

cp's OEIS Frontend

A000256 Number of simple triangulations of the plane with n nodes.

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A006390 Number of sensed loopless planar maps with n edges.

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A197271 a(n) = (10 / ((3*n+1)*(3*n+2))) * binomial(4*n, n).

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

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A002712 Number of unrooted triangulations of a disk that have reflection symmetry with n interior nodes and 3 nodes on the boundary.

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A007767 Number of pairs of permutations of degree n that avoid (12,21).

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A006391 Number of unsensed loopless planar maps with n edges.

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A197271 a(n) = (10 / ((3n+1)(3n+2))) binomial(4*n, n).