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.
1, 1, 1, 7, 6, 1, 73, 56, 16, 1, 879, 640, 208, 30, 1, 11713, 8256, 2848, 560, 48, 1, 167423, 115456, 41216, 9440, 1240, 70, 1, 2519937, 1710592, 624384, 156592, 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
Offset: 3
Examples
Triangle starts: n\k [2] [3] [4] [5] [6] [7] [8] [9] [10] [3] 1; [4] 1, 1; [5] 7, 6, 1; [6] 73, 56, 16, 1; [7] 879, 640, 208, 30, 1; [8] 11713, 8256, 2848, 560, 48, 1 [9] 167423, 115456, 41216, 9440, 1240, 70, 1; [10] 2519937, 1710592, 624384, 156592, 25864, 2408, 96, 1; [11] 39458047, 26468352, 9812992, 2613664, 496944, 61712, 4256, 126, 1; [12] ...
Links
- Gheorghe Coserea, Rows n=3..203, flattened
- M. Bodirsky, C. Groepl, D. Johannsen and M. Kang, A Direct Decomposition of 3-connected Planar Graphs, conference paper (FPSAC05).
Crossrefs
Programs
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PARI
x='x; y='y; system("wget http://oeis.org/A106651/a106651.txt"); Fy = read("a106651.txt"); A106651_ser(N) = { my(y0 = 1 + O(x^N), y1=0, n=1); while(n++, y1 = y0 - subst(Fy, y, y0)/subst(deriv(Fy, y), y, y0); if (y1 == y0, break()); y0 = y1); y0; }; z='z; t='t; u='u; c0='c0; r1 = 2*t*u + 2*t^2*u + 2*t*u^2 + 2*t^2*u^2; r2 = 4*t^2 + 4*t^3 + 4*t^2*u + 4*t^3*u; 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; 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; 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; r6 = u + 2*t*u + 2*t^2*u - t*u^2; Fz = r1*z^2 + (r3*c0 + r4)*z + r2*c0^2 + r5*c0 + r6; seq(N) = { N += 10; my(z0 = 1 + O(t^N) + O(u^N), z1=0, n=1, Fz = subst(Fz, 'c0, subst(A106651_ser(N), 'x, 't))); while(n++, z1 = z0 - subst(Fz, z, z0)/subst(deriv(Fz, z) , z, z0); if (z1 == z0, break()); z0 = z1); vector(N-10, n, vector(n, k, polcoeff(polcoeff(z0, n-k), k-1))); }; concat(seq(11))
Formula
A106651(n) = T(n,2) = Sum_{k=3..n-1} T(n,k), for n>=4.
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