A271238 - OEIS

%I #6 Apr 02 2016 21:33:25

%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 k-dimensional faces on a completely truncated n-dimensional simplex, 0 <= k <= n.

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

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

%F 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 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 (partially-truncated simplex).

%K nonn,tabl

%O 0,2

%A _Vincent J. Matsko_, Apr 02 2016