A271316 Triangle of numbers where T(n,k) is the number of k-dimensional faces on a partially truncated n-cube, 0 <= k <= n.

1, 2, 1, 8, 8, 1, 24, 36, 14, 1, 64, 128, 88, 24, 1, 160, 400, 400, 200, 42, 1, 384, 1152, 1520, 1120, 444, 76, 1, 896, 3136, 5152, 5040, 2968, 980, 142, 1, 2048, 8192, 16128, 19712, 15456, 7616, 2160, 272, 1, 4608, 20736, 47616, 69888, 68544, 45024, 19104, 4752, 530, 1, 10240, 51200, 134400, 230400, 271488, 223104, 126240, 47040, 10420, 1044, 1

Offset: 0

Author

Vincent J. Matsko, Apr 03 2016

Keywords

• nonn
• tabl

Examples

			Triangle begins:
1;
2, 1;
8, 8, 1;
24, 36, 14, 1;
64, 128, 88, 24, 1;
...
Row 2 describes an octagon: 8 vertices and 8 edges.
Row 3 describes a truncated cube: 24 vertices, 36 edges, and 14 faces.
		

Links

• Wikipedia, Truncated cube

Crossrefs

Cf. A038207 (n-cube).

Programs

Mathematica

Flatten[Table[
  CoefficientList[
   D[1 + Exp[(x + 2) z] + ( Exp[2 z (x + 1)] - (x + 1) Exp[2 z])/x, {z,
      k}] /. z -> 0, x], {k, 0, 10}]]

Formula

G.f. for rows (n > 0): (x+2)^n + 2^n*(x+1)*((x+1)^(n-1)-1)/x.

O.g.f: 1 + 1/(1-(x+2)*y) + 1/(x*(1-2*y*(x+1))) - (x+1)/(x*(1-2*y)).

E.g.f: 1 + exp((x+2)*z) + (exp(2*z*(x+1))-(x+1)*exp(2*z))/x.