A271238 Triangle of numbers where T(n,k) is the number of k-dimensional faces on a completely truncated n-dimensional simplex, 0 <= k <= n.

1, 2, 1, 3, 3, 1, 6, 12, 8, 1, 10, 30, 30, 10, 1, 15, 60, 80, 45, 12, 1, 21, 105, 175, 140, 63, 14, 1, 28, 168, 336, 350, 224, 84, 16, 1, 36, 252, 588, 756, 630, 336, 108, 18, 1, 45, 360, 960, 1470, 1512, 1050, 480, 135, 20, 1, 55, 495, 1485, 2640, 3234, 2772, 1650, 660, 165, 22, 1

Offset: 0

Author

Vincent J. Matsko, Apr 02 2016

Keywords

• nonn
• tabl

Comments

The complete truncation of a 1-dimensional segment is also a 1-dimensional segement (rather than degenerating to a point).

Examples

			Triangle begins:
1;
2, 1;
3, 3, 1;
6, 12, 8, 1;
10, 30, 30, 10, 1;
...
Row 2 describes a triangle.
Row 3 describes an octahedron.
		

Crossrefs

Cf. A259477 (partially-truncated simplex).

Programs

Mathematica

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

Formula

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

O.g.f.: (1/(1-(x+1)*y)^2-(x+1)/(1-y)^2)/x + 1/((1-(x+1)*y)*(1-y))+1+y*(x+1)*(1-1/(1-y)^3).

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