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

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

Author

Gheorghe Coserea, Aug 14 2017

Keywords

• nonn
• tabf

Comments

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

Examples

			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] ...
		

Links

• 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.

Crossrefs

Cf. A096332, A100960, A266390, A288265, A290326.

Programs

PARI

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))

Formula

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

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