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A288266 Triangle read by rows: T(n,k) is the number of labeled planar graphs on n vertices and k edges.

%I A288266 #11 Aug 15 2017 10:54:01
%S A288266 1,1,1,1,1,3,3,1,1,6,15,20,15,6,1,1,10,45,120,210,252,210,120,45,10,1,
%T A288266 15,105,455,1365,3003,5005,6435,6435,4995,2937,1125,195,1,21,210,1330,
%U A288266 5985,20349,54264,116280,203490,293860,351225,342405,255640,131985,40950,5712,1,28,378,3276,20475,98280,376740,1184040,3108105,6906620,13112694,21322812,29332947,32823084,28286520,17712016,7513632,1922760,223440
%N A288266 Triangle read by rows: T(n,k) is the number of labeled planar graphs on n vertices and k edges.
%C A288266 Row n >= 3 contains 3*n-5 terms.
%H A288266 Gheorghe Coserea, <a href="/A288266/b288266.txt">Rows n = 0..126, flattened</a>
%H A288266 E. A. Bender, Z. Gao and N. 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 A288266 M. Bodirsky, C. Groepl and M. 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 A288266 Omer Gimenez, Marc Noy, <a href="http://dx.doi.org/10.1090/S0894-0347-08-00624-3">Asymptotic enumeration and limit laws of planar graphs</a>, J. Amer. Math. Soc. 22 (2009), 309-329.
%F A288266 A066537(n) = Sum_{k=0..3*n-6} T(n,k) for n >= 3.
%F A288266 A007816(n-3) = T(n, 3*n-6).
%e A288266 A(x;t) = 1 + x + (1+t)*x^2/2! + (1+3*t+3*t^2+t^3)*x^3/3! + (1+6*t+15*t^2+20*t^3+15*t^4+6*t^5+t^6)*x^4/4! + ...
%e A288266 Triangle starts:
%e A288266 n\k [0] [1] [2]  [3]  [4]   [5]   [6]   [7]   [8]   [9]   [10]  [11]  [12]
%e A288266 [0] 1;
%e A288266 [1] 1;
%e A288266 [2] 1   1;
%e A288266 [3] 1,  3,  3,   1;
%e A288266 [4] 1,  6,  15,  20,  15,   6,    1;
%e A288266 [5] 1,  10, 45,  120, 210,  252,  210,  120,  45,   10;
%e A288266 [6] 1,  15, 105, 455, 1365, 3003, 5005, 6435, 6435, 4995, 2937, 1125, 195;
%e A288266 [7] ...
%o A288266 (PARI)
%o A288266 Q(n,k) = { \\ c-nets with n-edges, k-vertices
%o A288266   if (k < 2+(n+2)\3 || k > 2*n\3, return(0));
%o A288266   sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2*
%o A288266   (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) -
%o A288266   4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));
%o A288266 };
%o A288266 A100960_ser(N) = {
%o A288266 my(x='x+O('x^(3*N+1)), t='t+O('t^(N+4)),
%o A288266    q=t*x*Ser(vector(3*N+1, n, Polrev(vector(min(N+3, 2*n\3), k, Q(n,k)),'t))),
%o A288266    d=serreverse((1+x)/exp(q/(2*t^2*x) + t*x^2/(1+t*x))-1),
%o A288266    g2=intformal(t^2/2*((1+d)/(1+x)-1)));
%o A288266    serlaplace(Ser(vector(N, n, subst(polcoeff(g2, n,'t),'x,'t)))*'x);
%o A288266 };
%o A288266 A288266_seq(N) = {
%o A288266   my(x='x+O('x^(N+3)), b=t*x^2/2 + serconvol(A100960_ser(N), exp(x)),
%o A288266      g1=intformal(serreverse(x/exp(b'))/x));
%o A288266   apply(p->Vecrev(p), Vec(serlaplace(exp(g1))));
%o A288266 };
%o A288266 concat(A288266_seq(8))
%Y A288266 Cf. A096332, A100960, A266390, A288265, A290326.
%K A288266 nonn,tabf
%O A288266 0,6
%A A288266 _Gheorghe Coserea_, Aug 14 2017