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

Author

N. J. A. Sloane, Jan 12 2005

Keywords

• nonn
• tabf

Examples

			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] ...
		

Links

• Gheorghe Coserea, Rows n=3..126, flattened
• Edward A. Bender, Zhicheng Gao, and Nicholas C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.
• Manuel Bodirsky, Clemens Gröpl, and Mihyun Kang, Generating Labeled Planar Graphs Uniformly At Random, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, Pages 377-386.
• Jiaqi Liao, Hong Liu, and Guiying Yan, Isodiametric inequality for vector spaces, arXiv:2503.19239 [math.CO], 2025. See p. 8.

Crossrefs

Cf. A267411, A290326.

Row sums give A096331. Main diagonal is A001710.

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);
};
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

Extensions

More terms from Michel Marcus, Feb 10 2016