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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.

%I A271238 #6 Apr 02 2016 21:33:25
%S A271238 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 A271238 63,14,1,28,168,336,350,224,84,16,1,36,252,588,756,630,336,108,18,1,
%U A271238 45,360,960,1470,1512,1050,480,135,20,1,55,495,1485,2640,3234,2772,1650,660,165,22,1
%N 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.
%C A271238 The complete truncation of a 1-dimensional segment is also a 1-dimensional segement (rather than degenerating to a point).
%F A271238 G.f. for rows (n > 0): (((x+1)^n-1)*(x+n+2))/x-n-binomial(n+1,2)*(x+1).
%F A271238 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).
%F A271238 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 A271238 Triangle begins:
%e A271238 1;
%e A271238 2, 1;
%e A271238 3, 3, 1;
%e A271238 6, 12, 8, 1;
%e A271238 10, 30, 30, 10, 1;
%e A271238 ...
%e A271238 Row 2 describes a triangle.
%e A271238 Row 3 describes an octahedron.
%t A271238 Flatten[Table[
%t A271238   CoefficientList[
%t A271238    D[((x + 1) (z + 1) + 1) Exp[z] (Exp[x z] - 1)/x +
%t A271238       1 - (x + 1) z ((z + 2)*Exp[z] - 2)/2, {z, k}] /. z -> 0, x], {k, 0,
%t A271238    10}]]
%Y A271238 Cf. A259477 (partially-truncated simplex).
%K A271238 nonn,tabl
%O A271238 0,2
%A A271238 _Vincent J. Matsko_, Apr 02 2016