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

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

1,3

Comments

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

Links

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.

Formula

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).

Example

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] ...

Prog

(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);
};
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))

Crossrefs

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

Sequence in context: A048159 A276640 A123527 * A096611 A176666 A259031

Adjacent sequences: A288262 A288263 A288264 * A288266 A288267 A288268

Keyword

nonn,tabf

Author

Gheorghe Coserea, Aug 14 2017

Status

approved