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A285165 Triangle read by rows: T(n,k) is the number of c-nets with n-k inner vertices and k outer vertices, 3 <= n, 2 <= k <= n-1.

%I A285165 #25 Jul 28 2017 09:39:15
%S A285165 1,1,1,7,6,1,73,56,16,1,879,640,208,30,1,11713,8256,2848,560,48,1,
%T A285165 167423,115456,41216,9440,1240,70,1,2519937,1710592,624384,156592,
%U A285165 25864,2408,96,1,39458047,26468352,9812992,2613664,496944,61712,4256,126,1,637446145,423641088,158883840,44169600,9234368,1377600,132480,7008,160,1,10561615871,6966960128,2636197888,756712960,169378560,28663040,3430528,261648,10920,198,1
%N A285165 Triangle read by rows: T(n,k) is the number of c-nets with n-k inner vertices and k outer vertices, 3 <= n, 2 <= k <= n-1.
%H A285165 Gheorghe Coserea, <a href="/A285165/b285165.txt">Rows n=3..203, flattened</a>
%H A285165 M. Bodirsky, C. Groepl, D. Johannsen and M. Kang, <a href="http://www.emis.de/journals/SLC/wpapers/s54Abogrjoka.html">A Direct Decomposition of 3-connected Planar Graphs</a>, conference paper (FPSAC05).
%F A285165 A106651(n) = T(n,2) = Sum_{k=3..n-1} T(n,k), for n>=4.
%F A285165 T(n,n-2) = A054000(n-3) for n>= 5, T(n,n-3) = 8*A006325(n-3) for n>=6. - _Gheorghe Coserea_, Apr 19 2017
%e A285165 Triangle starts:
%e A285165 n\k  [2]       [3]       [4]      [5]      [6]     [7]    [8]   [9]  [10]
%e A285165 [3]  1;
%e A285165 [4]  1,        1;
%e A285165 [5]  7,        6,        1;
%e A285165 [6]  73,       56,       16,      1;
%e A285165 [7]  879,      640,      208,     30,      1;
%e A285165 [8]  11713,    8256,     2848,    560,     48,     1
%e A285165 [9]  167423,   115456,   41216,   9440,    1240,   70,    1;
%e A285165 [10] 2519937,  1710592,  624384,  156592,  25864,  2408,  96,   1;
%e A285165 [11] 39458047, 26468352, 9812992, 2613664, 496944, 61712, 4256, 126, 1;
%e A285165 [12] ...
%o A285165 (PARI)
%o A285165 x='x; y='y;
%o A285165 system("wget http://oeis.org/A106651/a106651.txt");
%o A285165 Fy = read("a106651.txt");
%o A285165 A106651_ser(N) = {
%o A285165   my(y0 = 1 + O(x^N), y1=0, n=1);
%o A285165   while(n++,
%o A285165     y1 = y0 - subst(Fy, y, y0)/subst(deriv(Fy, y), y, y0);
%o A285165     if (y1 == y0, break()); y0 = y1);
%o A285165   y0;
%o A285165 };
%o A285165 z='z; t='t; u='u; c0='c0;
%o A285165 r1 = 2*t*u + 2*t^2*u + 2*t*u^2 + 2*t^2*u^2;
%o A285165 r2 = 4*t^2 + 4*t^3 + 4*t^2*u + 4*t^3*u;
%o A285165 r3 = -4*t^2 - 4*t^3 - 2*t*u - 6*t^2*u - 4*t^3*u - 2*t*u^2 - 2*t^2*u^2;
%o A285165 r4 = 2*t + 2*t^2 + 4*t^3 - u + t*u + 4*t^3*u + u^2 + t*u^2 - 2*t^2*u^2;
%o A285165 r5 = -2*t - 2*t^2 - 4*t^3 - 4*t*u - 2*t^2*u - 4*t^3*u + 2*t^2*u^2;
%o A285165 r6 = u + 2*t*u + 2*t^2*u - t*u^2;
%o A285165 Fz = r1*z^2 + (r3*c0 + r4)*z + r2*c0^2 + r5*c0 + r6;
%o A285165 seq(N) = {
%o A285165   N += 10; my(z0 = 1 + O(t^N) + O(u^N), z1=0, n=1,
%o A285165   Fz = subst(Fz, 'c0, subst(A106651_ser(N), 'x, 't)));
%o A285165   while(n++,
%o A285165     z1 = z0 - subst(Fz, z, z0)/subst(deriv(Fz, z) , z, z0);
%o A285165     if (z1 == z0, break()); z0 = z1);
%o A285165   vector(N-10, n, vector(n, k, polcoeff(polcoeff(z0, n-k), k-1)));
%o A285165 };
%o A285165 concat(seq(11))
%Y A285165 Cf. A290326.
%Y A285165 Columns k=2-9 give: A106651(k=2), A285166(k=3), A285167(k=4), A285168(k=5), A285169(k=6), A285170(k=7), A285171(k=8), A285172(k=9).
%K A285165 nonn,tabl
%O A285165 3,4
%A A285165 _Gheorghe Coserea_, Apr 12 2017