%I
%S 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,
%T 63,14,1,28,168,336,350,224,84,16,1,36,252,588,756,630,336,108,18,1,
%U 45,360,960,1470,1512,1050,480,135,20,1,55,495,1485,2640,3234,2772,1650,660,165,22,1
%N Triangle of numbers where T(n,k) is the number of kdimensional faces on a completely truncated ndimensional simplex, 0 <= k <= n.
%C The complete truncation of a 1dimensional segment is also a 1dimensional segement (rather than degenerating to a point).
%F G.f. for rows (n > 0): (((x+1)^n1)*(x+n+2))/xnbinomial(n+1,2)*(x+1).
%F O.g.f.: (1/(1(x+1)*y)^2(x+1)/(1y)^2)/x + 1/((1(x+1)*y)*(1y))+1+y*(x+1)*(11/(1y)^3).
%F 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.
%e Triangle begins:
%e 1;
%e 2, 1;
%e 3, 3, 1;
%e 6, 12, 8, 1;
%e 10, 30, 30, 10, 1;
%e ...
%e Row 2 describes a triangle.
%e Row 3 describes an octahedron.
%t Flatten[Table[
%t CoefficientList[
%t D[((x + 1) (z + 1) + 1) Exp[z] (Exp[x z]  1)/x +
%t 1  (x + 1) z ((z + 2)*Exp[z]  2)/2, {z, k}] /. z > 0, x], {k, 0,
%t 10}]]
%Y Cf. A259477 (partiallytruncated simplex).
%K nonn,tabl
%O 0,2
%A _Vincent J. Matsko_, Apr 02 2016
