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

%I A343871 #11 Jan 02 2022 14:13:28
%S A343871 1,0,15,70,432,4320,30855,294840,2883240,28175952,310690800,
%T A343871 3458941920,40459730640,499638948480,6324655705200,83653192972800,
%U A343871 1145266802114400,16145338385736000,235579813593453000,3535776409508703360,54571687068401395200,866268656574795936000
%N A343871 Number of labeled 3-connected planar graphs with n edges.
%H A343871 Andrew Howroyd, <a href="/A343871/b343871.txt">Table of n, a(n) for n = 6..200</a>
%H A343871 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.
%F A343871 a(n) = Sum_{k=2+floor((n+2)/3)..floor(2*n/3)} k!*A290326(n-k+1, k-1)/(4*n).
%o A343871 (PARI)
%o A343871 Q(n,k) = { \\ c-nets with n-edges, k-vertices (see A290326)
%o A343871   if (k < 2+(n+2)\3 || k > 2*n\3, return(0));
%o A343871   sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2*
%o A343871   (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) -
%o A343871   4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));
%o A343871 };
%o A343871 a(n)={sum(k=2+(n+2)\3, 2*n\3, k!*Q(n,k))/(4*n)} \\ _Andrew Howroyd_, May 05 2021
%Y A343871 Cf. A000287, A002840 (unlabeled case), A096330, A290326, A291841, A338414.
%K A343871 nonn
%O A343871 6,3
%A A343871 _Andrew Howroyd_, May 05 2021