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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).

%I A100960 #42 Mar 28 2025 10:11:14
%S A100960 1,3,6,1,12,70,100,45,10,60,720,2445,3525,2637,1125,195,360,7560,
%T A100960 46830,132951,210861,205905,123795,40950,5712,2520,84000,835800,
%U A100960 3915240,10549168,18092368,20545920,15337560,7193760,1922760,223440,20160,997920,14757120,103692960,423918432,1119730032,2014030656,2516883516,2181661020,1285377660,491282820,109907280,10929600
%N 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).
%H A100960 Gheorghe Coserea, <a href="/A100960/b100960.txt">Rows n=3..126, flattened</a>
%H A100960 Edward A. Bender, Zhicheng Gao, and Nicholas C. Wormald, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v9i1r43">The number of labeled 2-connected planar graphs</a>, Electron. J. Combin., 9 (2002), #R43.
%H A100960 Manuel Bodirsky, Clemens Gröpl, and Mihyun Kang, <a href="http://dx.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, Pages 377-386.
%H A100960 Jiaqi Liao, Hong Liu, and Guiying Yan, <a href="https://arxiv.org/abs/2503.19239">Isodiametric inequality for vector spaces</a>, arXiv:2503.19239 [math.CO], 2025. See p. 8.
%e A100960 The triangle T(n,k), n>=3, k>=3 begins:
%e A100960   n\k [3] [4] [5] [6] [7]  [8]   [9]   [10]  [11]  [12]
%e A100960   [3] 1;
%e A100960   [4] 0,  3,  6,  1;
%e A100960   [5] 0,  0,  12, 70, 100, 45,   10;
%e A100960   [6] 0,  0,  0,  60, 720, 2445, 3525, 2637, 1125, 195;
%e A100960   [7] ...
%o A100960 (PARI)
%o A100960 Q(n,k) = { \\ c-nets with n-edges, k-vertices
%o A100960   if (k < 2+(n+2)\3 || k > 2*n\3, return(0));
%o A100960   sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2*
%o A100960   (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) -
%o A100960   4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));
%o A100960 };
%o A100960 A100960_ser(N) = {
%o A100960 my(x='x+O('x^(3*N+1)), t='t+O('t^(N+4)),
%o A100960    q=t*x*Ser(vector(3*N+1, n, Polrev(vector(min(N+3, 2*n\3), k, Q(n,k)),'t))),
%o A100960    d=serreverse((1+x)/exp(q/(2*t^2*x) + t*x^2/(1+t*x))-1),
%o A100960    g2=intformal(t^2/2*((1+d)/(1+x)-1)));
%o A100960    serlaplace(Ser(vector(N, n, subst(polcoeff(g2, n,'t),'x,'t)))*'x);
%o A100960 };
%o A100960 A100960_seq(N) = {
%o A100960   my(v=Vec(A100960_ser(N+2))); vector(#v, n, Vecrev(v[n]/t^(n+2)));
%o A100960 };
%o A100960 concat(A100960_seq(7)) \\ _Gheorghe Coserea_, Aug 09 2017
%Y A100960 Cf. A267411, A290326.
%Y A100960 Row sums give A096331. Main diagonal is A001710.
%K A100960 nonn,tabf
%O A100960 3,2
%A A100960 _N. J. A. Sloane_, Jan 12 2005
%E A100960 More terms from _Michel Marcus_, Feb 10 2016