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).

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

3,2

Links

Gheorghe Coserea, Rows n=3..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.

Example

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] ...

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);
};
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

Crossrefs

Cf. A267411, A290326.

Row sums give A096331. Main diagonal is A001710.

Sequence in context: A266151 A192100 A123534 * A130852 A228335 A138799

Adjacent sequences: A100957 A100958 A100959 * A100961 A100962 A100963

Keyword

nonn,tabf

Author

N. J. A. Sloane, Jan 12 2005

Extensions

More terms from Michel Marcus, Feb 10 2016

Status

approved