A290326 -id:A290326 - cp's OEIS frontend

A001508 a(n) is the number of c-nets with n+1 vertices and 2n+2 edges, n >= 1.

0, 0, 0, 0, 13, 252, 3740, 51300, 685419, 9095856, 120872850, 1614234960, 21697730835, 293695935764, 4003423965684, 54944523689692, 758990230992175, 10548884795729280, 147458773053913268, 2072369440050644208, 29271357456284966994

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..204
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+2,n)) \\ Gheorghe Coserea, Jul 28 2017

a(n) = A290326(n+2,n). - Gheorghe Coserea, Jul 28 2017

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

Name changed by Gheorghe Coserea, Jul 23 2017

A096330 Number of 3-connected planar graphs on n labeled nodes.

Original entry on oeis.org

1, 25, 1227, 84672, 7635120, 850626360, 112876089480, 17381709797760, 3046480841900160, 598731545755324800, 130389773403373545600, 31163616486434838067200, 8109213009296586130944000, 2282014010657773764160588800, 690521215428258768326957184000

Offset: 4

Steven Finch, Aug 02 2004

nonn

Recurrence known, see Bodirsky et al.

M. Bodirsky, C. Groepl and M. Kang: Generating Labeled Planar Graphs Uniformly At Random; ICALP03 Eindhoven, LNCS 2719, Springer Verlag (2003), 1095 - 1107.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 419.

Gheorghe Coserea, Table of n, a(n) for n = 4..104
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, pp. 377-386.

Cf. A066537, A096331, A096332, 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))));
};
a(n) = sum(k=(3*n+1)\2, 3*n-6, n!*Q(k,n)/(4*k));
apply(a, [4..18]) \\ Gheorghe Coserea, Aug 11 2017

A106651 c(n) = number of c-nets on n vertices.

Original entry on oeis.org

1, 1, 7, 73, 879, 11713, 167423, 2519937, 39458047, 637446145, 10561615871, 178683815937, 3076487458815, 53766284722177, 951817354412031, 17039752595865601, 308068940431556607, 5618467344224354305

Offset: 3

Daniel Johannsen (johannse(AT)informatik.hu-berlin.de), May 12 2005

Definition of c-net: a 3-connected planar map, rooted by a directed edge on the outer face.

			c(0)=c(1)=1 because the only c-nets on 3 respectively 4 vertices are the complete graphs.

Gheorghe Coserea, Table of n, a(n) for n = 3..302
M. Bodirsky, C. Groepl, D. Johannsen and M. Kang, A Direct Decomposition of 3-connected Planar Graphs, conference paper (FPSAC05).
Gheorghe Coserea, Algebraic equation for g.f.
R. C. Mullin, P. J. Schellenberg, The enumeration of c-nets via quadrangulations, J. Combinatorial Theory 4 1968 259--276. MR0218275 (36 #1362).

Cf. A001506, A001507, A001508, A285165, A290326.

c[0] = 1; c[1] = 1; c[2] = 7; c[3] = 73; c[4] = 879; c[5] = 11713; c[6] = 167423; c[7] = 2519937; c[n_] := c[n] = ( (-189665280 + 134270976 n - 31309824 n^2 + 2408448 n^3) c[n - 7] + (-479162880 + 376680448 n - 98932224 n^2 + 8692736 n^3) c[n - 6] + (-446660160 + 384601888 n - 112131264 n^2 + 11026784 n^3) c[n - 5] + (-183645792 + 168826836 n - 52598160 n^2 + 5361276 n^3) c[n - 4] + (-25324080 + 24563948 n - 6853668 n^2 + 418816 n^3) c[n - 3] + (1156086 - 2064937 n + 1206966 n^2 - 180467 n^3) c[n - 2] + (-3192 + 4842 n - 29796 n^2 + 18930 n^3) c[n - 1] ) / (126 + 693 n + 1134 n^2 + 567 n^3);

x='x; y='y;
system("wget http://oeis.org/A106651/a106651.txt");
Fxy = read("a106651.txt");
seq(N) = {
  my(y0 = 1 + O('x^N), y1=0);
  for (k = 1, N,
    y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);
    if (y1 == y0, break()); y0 = y1);
  Vec(y0);
};
seq(18)  \\ Gheorghe Coserea, Jan 08 2017

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))));
};
a(n) = if (n==3, 1, sum(k = (n+3)\2, 2*n-5, A290326(n-1, k)));
vector(18, n, a(n+2)) \\ Gheorghe Coserea, Jul 28 2017

c(0)=1, c(1) = 1, c(2) = 7, c(3) = 73, c(4) = 879, c(5) = 11713, c(6) = 167423, c(7) = 2519937, c(n) = ( (-189665280 + 134270976 n - 31309824 n^2 + 2408448 n^3) c(n-7) + (-479162880 + 376680448 n - 98932224 n^2 + 8692736 n^3) c(n-6) + (-446660160 + 384601888 n - 112131264 n^2 + 11026784 n^3) c(n-5) + (-183645792 + 168826836 n - 52598160 n^2 + 5361276 n^3) c(n-4) + (-25324080 + 24563948 n - 6853668 n^2 + 418816 n^3) c(n-3) + (1156086 - 2064937 n + 1206966 n^2 - 180467 n^3) c(n-2) + (-3192 + 4842 n - 29796 n^2 + 18930 n^3) c(n-1) ) / (126 + 693 n + 1134 n^2 + 567 n^3). Generating function C(t)=sum_(n>=0){c(n-3)t^n} implicitly given by: 0 = -1 + C(t) + 36 t - 43 C(t) t + 6 C(t)^2 t + 131 t^2 - 337 C(t) t^2 + 218 C(t)^2 t^2 + 12 C(t)^3 t^2 + 350 t^3 - 1021 C(t) t^3 + 894 C(t)^2 t^3 - 228 C(t)^3 t^3 + 8 C(t)^4 t^3 + 540 t^4 - 1828 C(t) t^4 + 2190 C(t)^2 t^4 - 988 C(t)^3 t^4 + 72 C(t)^4 t^4 + 616 t^5 - 2404 C(t) t^5 + 3284 C(t)^2 t^5 - 1756 C(t)^3 t^5 + 264 C(t)^4 t^ 5 + 536 t^6 - 2128 C(t) t^6 + 3120 C(t)^2 t^6 - 2032 C(t)^3 t^6 + 504 C(t)^4 t^6 + 304 t^7 - 1344 C(t) t^7 + 2304 C(t)^2 t^7 - 1792 C(t)^3 t^7 + 528 C(t)^4 t^7 + 160 t^8 - 768 C(t) t^8 + 1344 C(t)^2 t^8 - 1024 C(t)^3 t^8 + 288 C(t)^4 t^8 + 64 t^9 - 256 C(t) t^9 + 384 C(t)^2 t^9 - 256 C(t)^3 t^9 + 64 C(t)^4 t^9. Explicit generating function can be obtained using Mathematica.

Mathematica code improved by David Radcliffe, Feb 12 2011

A318101 Number of rooted 2-connected 4-regular planar maps, which may have loops, with n inner faces.

Original entry on oeis.org

2, 9, 30, 154, 986, 6977, 52590, 415678, 3409032, 28787498, 248930292, 2195238596, 19682012382, 178974809121, 1647460326046, 15327261314934, 143942130406288, 1363094805806462, 13004498819335396, 124900418475706476, 1206861624598185332, 11725558427958257690, 114494070652568918380

Offset: 2

Gheorghe Coserea, Aug 16 2018

nonn

			A(x) = 2*x^2 + 9*x^3 + 30*x^4 + 154*x^5 + 986*x^6 + 6977*x^7 + 52590*x^8 + ...

Gheorghe Coserea, Table of n, a(n) for n = 2..307
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.

Cf. A058860, A099553, A290326, A318102, A318103.

F = (1 - z + 2*z^3*x*(1 - x))/(2*z*(1 - z^2*x));
G = x*(4*x^2 + 1)*z^4 - 4*x*(2*x - 1)*z^3 - 5*x*z^2 + 2*(2*x - 1)*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((1 + 2*x)*subst(F, 'z, Z(N+2)));
seq(23)
\\ test: y=Ser(seq(303))*x^2; 0 == 8*y^4 + 4*(2*x + 1)*(2*x + 3)*y^3 - (x - 6)*(2*x + 1)^2*y^2 + (2*x + 1)^3*(18*x^2 - 16*x + 1)*y + x^2*(2*x + 1)^4*(27*x - 2)

G.f.: (1 + 2*x)*F, where F = (1 - z + 2*z^3*x*(1 - x))/(2*z*(1 - z^2*x)) and z = 1 + 2*x + 6*x^2 + 34*x^3 + 254*x^4 + 2052*x^5 + 17332*x^6 + 151658*x^7 + ... satisfies 0 = x*(4*x^2 + 1)*z^4 - 4*x*(2*x - 1)*z^3 - 5*x*z^2 + 2*(2*x - 1)*z + 2. (see Theorem C in link)
G.f. y=A(x) satisfies:
0 = 8*y^4 + 4*(2*x + 1)*(2*x + 3)*y^3 - (x - 6)*(2*x + 1)^2*y^2 + (2*x + 1)^3*(18*x^2 - 16*x + 1)*y + x^2*(2*x + 1)^4*(27*x - 2).
0 = x^3*(2*x + 1)^4*(4*x^2 - 2*x + 1)*(108*x^2 - 304*x + 27)*(128*x^6 - 1504*x^5 + 5864*x^4 - 8282*x^3 + 4381*x^2 - 659*x + 60)*y'''' - x^2*(2*x + 1)^3*(417792*x^10 - 1973504*x^9 - 7891840*x^8 + 53958576*x^7 - 106786208*x^6 + 92663096*x^5 - 38721768*x^4 + 9604075*x^3 - 1447438*x^2 + 141966*x - 4860)*y''' + 3*x*(2*x + 1)^2*(163840*x^12 - 1929216*x^11 + 11348480*x^10 - 47888896*x^9 + 125855008*x^8 - 184580160*x^7 + 158611640*x^6 - 81013580*x^5 + 22892592*x^4 - 3821021*x^3 + 403960*x^2 - 23876*x + 120)*y'' - 12*(2*x + 1)*(163840*x^13 - 1888256*x^12 + 11294208*x^11 - 48430080*x^10 + 125093344*x^9 - 184709184*x^8 + 159190952*x^7 - 80413964*x^6 + 23140740*x^5 - 3792653*x^4 + 391233*x^3 - 28410*x^2 - 199*x + 30)*y' + 24*(163840*x^13 - 1847296*x^12 + 11198976*x^11 - 48855552*x^10 + 124699296*x^9 - 184627968*x^8 + 159583928*x^7 - 80114156*x^6 + 23238984*x^5 - 3787577*x^4 + 385076*x^3 - 30072*x^2 - 292*x + 40)*y.
a(n) ~ c / (sqrt(Pi) * n^(5/2) * r^n), where r = (76 - 7*sqrt(103))/54 and c = sqrt(3278181/(3125*(109592 + 10823*sqrt(103)))). - Vaclav Kotesovec, Aug 25 2018

A318102 Number of rooted 2-connected 4-regular maps on the projective plane, which may have loops, with n inner faces.

Original entry on oeis.org

5, 38, 199, 1466, 12365, 109700, 1003929, 9404402, 89690920, 867506788, 8486154214, 83790178300, 833805753167, 8352569222312, 84150924820499, 852039732062530, 8664839058268872, 88459350543053228, 906208005777385526, 9312350891307447116, 95963703215086597466, 991421114632619679480

Offset: 1

Gheorghe Coserea, Aug 19 2018

nonn

			A(x) = 5*x + 38*x^2 + 199*x^3 + 1466*x^4 + 12365*x^5 + 109700*x^6 + ...

Gheorghe Coserea, Table of n, a(n) for n = 1..303
Shude Long, Han Ren, Counting 2-Connected 4-Regular Maps on the Projective Plane, Volume 21, Issue 2 (2014), Paper #P2.51.
Gheorghe Coserea, Annihilating differential operator

A058860, A099553, A290326, A318101, A318103.

F = (1 - z + 2*z^3*x*(1 - x))/(2*z*(1 - z^2*x));
G = x*(4*x^2 + 1)*z^4 - 4*x*(2*x - 1)*z^3 - 5*x*z^2 + 2*(2*x - 1)*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;
};
f(N) = subst((z + 2*z*F - 1)/(2*z^2*x), 'z, Z(N));
Fp2(N) = {
  my(z=Z(N), f=f(N));
  ((1 - sqrt(1 - 4*z^2*x*f))/(x*z) + z*(z-3)/2 * f^2 + 2*(f - 1))/(2*f - 1);
};
Fp4(N) = (1 + 2*x)*(Fp2(N) - 1) + 3*subst(F, 'z, Z(N+2));
seq(N) = Vec(Fp4(N+1));
seq(22)
/* test:
system("wget https://oeis.org/A318102/a318102.txt");
apply_diffop(p, s) = { \\ apply diffop p (encoded as Pol in Dx) to Ser s
  s=intformal(s);
  sum(n=0, poldegree(p, 'Dx), s=s'; polcoeff(p, n, 'Dx) * s);
};
0 == apply_diffop(read("a318102.txt"), Fp4(1001))
*/

G.f.: (1 + 2*x)*(Fp2 - 1) + 3*F, where Fp2 and F are given by the system of algebraic equations:
0 = x*(4*x^2 + 1)*z^4 - 4*x*(2*x - 1)*z^3 - 5*x*z^2 + 2*(2*x - 1)*z + 2,
F = (1 - z + 2*z^3*x*(1 - x))/(2*z*(1 - z^2*x)),
f = (z + 2*z*F - 1)/(2*z^2*x),
Fp2 = ((1 - sqrt(1 - 4*z^2*x*f))/(x*z) + z*(z-3)/2 * f^2 + 2*(f - 1))/(2*f - 1).
The initial coefficients of the solutions are:
z = 1 + 2*x + 6*x^2 + 34*x^3 + 254*x^4 + 2052*x^5 + 17332*x^6 + ...,
F = 2*x^2 + 5*x^3 + 20*x^4 + 114*x^5 + 758*x^6 + 5461*x^7 + 41668*x^8 + ...,
f = 1 + x + 6*x^2 + 37*x^3 + 262*x^4 + 2050*x^5 + 17064*x^6 + ...,
Fp2 = 1 + 5*x + 22*x^2 + 140*x^3 + 1126*x^4 + 9771*x^5 + 87884*x^6 + ...
(see Facts 2-5 and Theorem B in the link)
G.f. y=A(x) satisfies:
0 = 4096*x^7*(4*x^2 - 2*x + 1)^2*y^8 + 2048*x^6*(4*x^2 - 2*x + 1)^2*(36*x^2 + 16*x - 7)*y^7 + 128*x^5*(4*x^2 - 2*x + 1)*(8320*x^6 + 19648*x^5 - 1076*x^4 - 1804*x^3 + 1907*x^2 - 580*x + 126)*y^6 + 32*x^4*(4*x^2 - 2*x + 1)*(225280*x^7 + 444240*x^6 + 84688*x^5 - 29552*x^4 + 32044*x^3 - 9577*x^2 + 976*x - 280)*y^5 + x^3*(40239104*x^11 - 79837184*x^10 + 295013376*x^9 - 58917488*x^8 + 30598624*x^7 + 31536856*x^6 - 14288200*x^5 + 7449849*x^4 - 1791392*x^3 + 303304*x^2 - 15680*x + 2800)*y^4 + 2*x^2*(272629760*x^13 - 24282112*x^12 - 175736320*x^11 + 322666592*x^10 - 42540704*x^9 + 44400384*x^8 + 30919616*x^7 - 7960626*x^6 + 8259482*x^5 - 2256409*x^4 + 613344*x^3 - 92803*x^2 + 2512*x - 252)*y^3 + x*(4137222144*x^14 + 1879746560*x^13 - 1113429024*x^12 + 1342878720*x^11 + 65189712*x^10 + 10079664*x^9 + 147999470*x^8 - 45142196*x^7 + 25711384*x^6 - 6520084*x^5 + 2042177*x^4 - 392900*x^3 + 48476*x^2 - 1064*x + 49)*y^2 - 2*(1128267776*x^16 - 4335727616*x^15 - 6678567648*x^14 + 1061181280*x^13 - 2785972352*x^12 + 213096160*x^11 - 166061526*x^10 - 112334126*x^9 + 50212017*x^8 - 27194278*x^7 + 7091863*x^6 - 1701882*x^5 + 350358*x^4 - 36314*x^3 + 2951*x^2 - 44*x + 1)*y + x*(17448304640*x^16 - 38432538624*x^15 + 29298729744*x^14 - 1261398240*x^13 + 9372670936*x^12 + 6841726488*x^11 + 1476038993*x^10 + 1644370884*x^9 + 177903076*x^8 + 98892200*x^7 + 15461596*x^6 - 2656592*x^5 + 901090*x^4 - 145464*x^3 + 25339*x^2 - 364*x + 10).

A318103 Number of rooted 2-connected loopless 4-regular maps on the projective plane with n inner faces.

Original entry on oeis.org

6, 21, 138, 781, 4836, 30099, 191698, 1236024, 8063492, 53086930, 352249244, 2352800079, 15805224904, 106702428453, 723509453442, 4924851788720, 33638721268140, 230477992427450, 1583550831926508, 10907729315809642, 75307599054762424, 521026923863915206, 3611800088179535100

Offset: 2

Gheorghe Coserea, Aug 20 2018

nonn

			A(x) = 6*x^2 + 21*x^3 + 138*x^4 + 781*x^5 + 4836*x^6 + 30099*x^7 + ...

Gheorghe Coserea, Table of n, a(n) for n = 2..304
Shude Long, Han Ren, Counting 2-Connected 4-Regular Maps on the Projective Plane, Volume 21, Issue 2 (2014), Paper #P2.51.
Gheorghe Coserea, Annihilating differential operator

Cf. A058860, A099553, A290326, A318101, A318102.

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;
};
f(N) = subst((z - 1 + 2*z*x + 2*z*F)/(2*x*z^2), 'z, Z(N));
Fp4(N) = {
  my(z=Z(N), f=f(N));
((1 - sqrt(1- 4*x*z^2*f))/(x*z) - z*f - x*(z*f)^3*(2 - f))/(2*f - 1) - 1;
};
seq(N) = Vec(Fp4(N+2));
seq(23)
/* test:
system("wget https://oeis.org/A318103/a318103.txt");
apply_diffop(p, s) = {
  s=intformal(s);
  sum(n=0, poldegree(p, 'Dx), s=s'; polcoeff(p, n, 'Dx) * s);
};
0 == apply_diffop(read("a318103.txt"), Fp4(1001))
*/

G.f.: ((1 - sqrt(1- 4*x*z^2*f))/(x*z) - z*f - x*(z*f)^3*(2 - f))/(2*f - 1) - 1, where z and f are given by the system of algebraic equations:
0 = x*(4*x + 1)*z^4 + 4*x*z^3 - 5*x*z^2 - 2*z + 2,
F = (2*z^3*x^2 + (2*z^3 - 2*z)*x + (-z + 1))/(-2*z^3*x + 2*z),
f = (z - 1 + 2*z*x + 2*z*F)/(2*x*z^2).
The initial coefficients of the solutions are:
z = 1 + 2*x^2 + 6*x^3 + 34*x^4 + 176*x^5 + 1004*x^6 + 5858*x^7 + ...
F = x^3 + 2*x^4 + 10*x^5 + 42*x^6 + 209*x^7 + 1066*x^8 + 5726*x^9 + ...
f = 1 + x + 2*x^2 + 9*x^3 + 42*x^4 + 222*x^5 + 1232*x^6 + 7137*x^7 + ...
(see Facts 6-7 and Theorem C in the link)
G.f. y=A(x) satisfies:
0 = 4096*x^7*(2*x + 1)^2*y^8 + 2048*x^6*(2*x + 1)^2*(16*x - 7)*y^7 + 128*x^5*(2*x + 1)*(1792*x^3 - 285*x^2 - 76*x + 126)*y^6 + 32*x^4*(2*x + 1)*(14336*x^4 - 2360*x^3 - 57*x^2 - 144*x - 280)*y^5 + x^3*(1146880*x^6 + 625920*x^5 + 282633*x^4 + 174368*x^3 + 44232*x^2 + 6720*x + 2800)*y^4 + 2*x^2*(2*x + 1)*(229376*x^6 + 108288*x^5 + 419113*x^4 + 53390*x^3 - 39619*x^2 + 1000*x - 252)*y^3 + x*(458752*x^8 + 740608*x^7 + 3399862*x^6 + 1371564*x^5 - 317093*x^4 - 58308*x^3 + 25400*x^2 - 672*x + 49)*y^2 + 2*(65536*x^9 + 162048*x^8 + 1258098*x^7 + 287981*x^6 - 86682*x^5 + 22504*x^4 + 5250*x^3 - 2026*x^2 + 36*x - 1)*y + x^2*(16384*x^7 + 58112*x^6 + 674825*x^5 + 33912*x^4 + 11954*x^3 + 23076*x^2 - 390*x + 12).
From Vaclav Kotesovec, Aug 25 2018: (Start)
a(n) ~ c1 * (196/27)^n / n^(5/4) * (1 + c2/n^(1/4) + c3/n^(1/2)), where
c1 = 7^(5/4) * Gamma(1/4) / (5^(5/4) * 3^(3/4) * Pi),
c2 = -17 * 7^(1/4) * sqrt(Pi) / (3^(7/4) * 5^(1/4) * Gamma(1/4)),
c3 = 71 * sqrt(7) * Pi / (2^(3/2) * sqrt(3) * 5^(3/2) * Gamma(1/4)^2). (End)

A288266 Triangle read by rows: T(n,k) is the number of labeled planar graphs on n vertices and k edges.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 1, 6, 15, 20, 15, 6, 1, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 4995, 2937, 1125, 195, 1, 21, 210, 1330, 5985, 20349, 54264, 116280, 203490, 293860, 351225, 342405, 255640, 131985, 40950, 5712, 1, 28, 378, 3276, 20475, 98280, 376740, 1184040, 3108105, 6906620, 13112694, 21322812, 29332947, 32823084, 28286520, 17712016, 7513632, 1922760, 223440

Offset: 0

Gheorghe Coserea, Aug 14 2017

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

			A(x;t) = 1 + x + (1+t)*x^2/2! + (1+3*t+3*t^2+t^3)*x^3/3! + (1+6*t+15*t^2+20*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]
[0] 1;
[1] 1;
[2] 1   1;
[3] 1,  3,  3,   1;
[4] 1,  6,  15,  20,  15,   6,    1;
[5] 1,  10, 45,  120, 210,  252,  210,  120,  45,   10;
[6] 1,  15, 105, 455, 1365, 3003, 5005, 6435, 6435, 4995, 2937, 1125, 195;
[7] ...

Gheorghe Coserea, Rows n = 0..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. A096332, A100960, A266390, A288265, 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);
};
A288266_seq(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));
  apply(p->Vecrev(p), Vec(serlaplace(exp(g1))));
};
concat(A288266_seq(8))

A066537(n) = Sum_{k=0..3*n-6} T(n,k) for n >= 3.

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

A343871 Number of labeled 3-connected planar graphs with n edges.

Original entry on oeis.org

1, 0, 15, 70, 432, 4320, 30855, 294840, 2883240, 28175952, 310690800, 3458941920, 40459730640, 499638948480, 6324655705200, 83653192972800, 1145266802114400, 16145338385736000, 235579813593453000, 3535776409508703360, 54571687068401395200, 866268656574795936000

Offset: 6

Andrew Howroyd, May 05 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 6..200
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, pp. 377-386.

Cf. A000287, A002840 (unlabeled case), A096330, A290326, A291841, A338414.

Q(n,k) = { \\ c-nets with n-edges, k-vertices (see A290326)
  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))));
};
a(n)={sum(k=2+(n+2)\3, 2*n\3, k!*Q(n,k))/(4*n)} \\ Andrew Howroyd, May 05 2021

a(n) = Sum_{k=2+floor((n+2)/3)..floor(2*n/3)} k!*A290326(n-k+1, k-1)/(4*n).

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