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

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

Offset: 6

Author

Andrew Howroyd, May 05 2021

Keywords

• nonn

Links

• 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.

Crossrefs

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

Programs

PARI

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

Formula

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