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

Author

Gheorghe Coserea, Apr 12 2017

Keywords

• nonn
• tabl

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

Cf. A290326.

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).

Programs

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