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

%I A271316 #9 Apr 05 2016 01:02:59
%S A271316 1,2,1,8,8,1,24,36,14,1,64,128,88,24,1,160,400,400,200,42,1,384,1152,
%T A271316 1520,1120,444,76,1,896,3136,5152,5040,2968,980,142,1,2048,8192,16128,
%U A271316 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
%N A271316 Triangle of numbers where T(n,k) is the number of k-dimensional faces on a partially truncated n-cube, 0 <= k <= n.
%H A271316 Wikipedia, <a href="https://en.wikipedia.org/wiki/Truncated_cube">Truncated cube</a>
%F A271316 G.f. for rows (n > 0): (x+2)^n + 2^n*(x+1)*((x+1)^(n-1)-1)/x.
%F A271316 O.g.f: 1 + 1/(1-(x+2)*y) + 1/(x*(1-2*y*(x+1))) - (x+1)/(x*(1-2*y)).
%F A271316 E.g.f: 1 + exp((x+2)*z) + (exp(2*z*(x+1))-(x+1)*exp(2*z))/x.
%e A271316 Triangle begins:
%e A271316 1;
%e A271316 2, 1;
%e A271316 8, 8, 1;
%e A271316 24, 36, 14, 1;
%e A271316 64, 128, 88, 24, 1;
%e A271316 ...
%e A271316 Row 2 describes an octagon: 8 vertices and 8 edges.
%e A271316 Row 3 describes a truncated cube: 24 vertices, 36 edges, and 14 faces.
%t A271316 Flatten[Table[
%t A271316   CoefficientList[
%t A271316    D[1 + Exp[(x + 2) z] + ( Exp[2 z (x + 1)] - (x + 1) Exp[2 z])/x, {z,
%t A271316       k}] /. z -> 0, x], {k, 0, 10}]]
%Y A271316 Cf. A038207 (n-cube).
%K A271316 nonn,tabl
%O A271316 0,2
%A A271316 _Vincent J. Matsko_, Apr 03 2016