A096330 - OEIS

%I #21 Dec 25 2018 22:35:58

%S 1,25,1227,84672,7635120,850626360,112876089480,17381709797760,

%T 3046480841900160,598731545755324800,130389773403373545600,

%U 31163616486434838067200,8109213009296586130944000,2282014010657773764160588800,690521215428258768326957184000

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

%C Recurrence known, see Bodirsky et al.

%D M. Bodirsky, C. Groepl and M. Kang: Generating Labeled Planar Graphs Uniformly At Random; ICALP03 Eindhoven, LNCS 2719, Springer Verlag (2003), 1095 - 1107.

%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 419.

%H Gheorghe Coserea, <a href="/A096330/b096330.txt">Table of n, a(n) for n = 4..104</a>

%H M. Bodirsky, C. Groepl and M. Kang, <a href="https://doi.org/10.1016/j.tcs.2007.02.045">Generating Labeled Planar Graphs Uniformly At Random</a>, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, pp. 377-386.

%o (PARI)

%o Q(n,k) = { \\ c-nets with n-edges, k-vertices

%o   if (k < 2+(n+2)\3 || k > 2*n\3, return(0));

%o   sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2*

%o   (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) -

%o   4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));

%o };

%o a(n) = sum(k=(3*n+1)\2, 3*n-6, n!*Q(k,n)/(4*k));

%o apply(a, [4..18]) \\ _Gheorghe Coserea_, Aug 11 2017

%Y Cf. A066537, A096331, A096332, A290326.

%K nonn

%O 4,2

%A _Steven Finch_, Aug 02 2004