A096330 Number of 3-connected planar graphs on n labeled nodes.

1, 25, 1227, 84672, 7635120, 850626360, 112876089480, 17381709797760, 3046480841900160, 598731545755324800, 130389773403373545600, 31163616486434838067200, 8109213009296586130944000, 2282014010657773764160588800, 690521215428258768326957184000

Offset: 4

Author

Steven Finch, Aug 02 2004

Keywords

• nonn

Comments

Recurrence known, see Bodirsky et al.

References

• M. Bodirsky, C. Groepl and M. Kang: Generating Labeled Planar Graphs Uniformly At Random; ICALP03 Eindhoven, LNCS 2719, Springer Verlag (2003), 1095 - 1107.
• Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 419.

Links

• Gheorghe Coserea, Table of n, a(n) for n = 4..104
• 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. A066537, A096331, A096332, 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))));
};
a(n) = sum(k=(3*n+1)\2, 3*n-6, n!*Q(k,n)/(4*k));
apply(a, [4..18]) \\ Gheorghe Coserea, Aug 11 2017