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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.
9

%I #25 Jul 28 2017 09:39:15

%S 1,1,1,7,6,1,73,56,16,1,879,640,208,30,1,11713,8256,2848,560,48,1,

%T 167423,115456,41216,9440,1240,70,1,2519937,1710592,624384,156592,

%U 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 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 Gheorghe Coserea, <a href="/A285165/b285165.txt">Rows n=3..203, flattened</a>

%H 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 A106651(n) = T(n,2) = Sum_{k=3..n-1} T(n,k), for n>=4.

%F 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 Triangle starts:

%e n\k [2] [3] [4] [5] [6] [7] [8] [9] [10]

%e [3] 1;

%e [4] 1, 1;

%e [5] 7, 6, 1;

%e [6] 73, 56, 16, 1;

%e [7] 879, 640, 208, 30, 1;

%e [8] 11713, 8256, 2848, 560, 48, 1

%e [9] 167423, 115456, 41216, 9440, 1240, 70, 1;

%e [10] 2519937, 1710592, 624384, 156592, 25864, 2408, 96, 1;

%e [11] 39458047, 26468352, 9812992, 2613664, 496944, 61712, 4256, 126, 1;

%e [12] ...

%o (PARI)

%o x='x; y='y;

%o system("wget http://oeis.org/A106651/a106651.txt");

%o Fy = read("a106651.txt");

%o A106651_ser(N) = {

%o my(y0 = 1 + O(x^N), y1=0, n=1);

%o while(n++,

%o y1 = y0 - subst(Fy, y, y0)/subst(deriv(Fy, y), y, y0);

%o if (y1 == y0, break()); y0 = y1);

%o y0;

%o };

%o z='z; t='t; u='u; c0='c0;

%o r1 = 2*t*u + 2*t^2*u + 2*t*u^2 + 2*t^2*u^2;

%o r2 = 4*t^2 + 4*t^3 + 4*t^2*u + 4*t^3*u;

%o 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 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 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 r6 = u + 2*t*u + 2*t^2*u - t*u^2;

%o Fz = r1*z^2 + (r3*c0 + r4)*z + r2*c0^2 + r5*c0 + r6;

%o seq(N) = {

%o N += 10; my(z0 = 1 + O(t^N) + O(u^N), z1=0, n=1,

%o Fz = subst(Fz, 'c0, subst(A106651_ser(N), 'x, 't)));

%o while(n++,

%o z1 = z0 - subst(Fz, z, z0)/subst(deriv(Fz, z) , z, z0);

%o if (z1 == z0, break()); z0 = z1);

%o vector(N-10, n, vector(n, k, polcoeff(polcoeff(z0, n-k), k-1)));

%o };

%o concat(seq(11))

%Y Cf. A290326.

%Y 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 nonn,tabl

%O 3,4

%A _Gheorghe Coserea_, Apr 12 2017