cp's OEIS Frontend

A259477 Triangle of numbers where T(n,k) is the number of k-dimensional faces on a partially truncated n-dimensional simplex, 0 <= k <= n.

%I A259477 #35 Apr 02 2016 19:50:00
%S A259477 1,2,1,6,6,1,12,18,8,1,20,40,30,10,1,30,75,80,45,12,1,42,126,175,140,
%T A259477 63,14,1,56,196,336,350,224,84,16,1,72,288,588,756,630,336,108,18,1,
%U A259477 90,405,960,1470,1512,1050,480,135,20,1,110,550,1485,2640,3234,2772,1650,660,165,22,1
%N A259477 Triangle of numbers where T(n,k) is the number of k-dimensional faces on a partially truncated n-dimensional simplex, 0 <= k <= n.
%H A259477 Charles R Greathouse IV, <a href="/A259477/b259477.txt">Table of n, a(n) for n = 0..10000</a>
%F A259477 T(n,0) = n*(n+1), n > 0; T(n,k) = (n+1-k)*binomial(n+1,k+1), 1 <= k <= n.
%F A259477 E.g.f.: ((x+1)*(z+1)+1)*exp(z)*(exp(x*z)-1)/x + 1.
%F A259477 From _Vincent J. Matsko_, Jul 18 2015: (Start)
%F A259477 O.g.f.: (1/(1-(x+1)*y)^2-(x+1)/(1-y)^2)/x + 1/((1-(x+1)y)*(1-y))+1.
%F A259477 G.f. for rows (n > 0): (((x+1)^n-1)*(x+n+2))/x-n. (End)
%e A259477 Triangle begins:
%e A259477 1;
%e A259477 2, 1;
%e A259477 6, 6, 1;
%e A259477 12, 18, 8, 1;
%e A259477 20, 40, 30, 10, 1;
%e A259477 ...
%t A259477 Join @@ (CoefficientList[#,
%t A259477      x] & /@ (Expand[
%t A259477        D[((x + 1) (z + 1) + 1) Exp[z] (Exp[x z] - 1)/x + 1, {z, #}] /.
%t A259477         z -> 0] & /@ Range[0, 10])) (* _Vincent J. Matsko_, Jun 30 2015 *)
%t A259477 Flatten[Table[
%t A259477   CoefficientList[
%t A259477    D[(1/(1 - (x + 1)*y)^2 - (x + 1)/(1 - y)^2)/x +
%t A259477        1/((1 - (x + 1) y)*(1 - y)) + 1, {y, k}]/Factorial[k] /. y -> 0,
%t A259477    x], {k, 0, 10}]] (* _Vincent J. Matsko_, Jul 18 2015 *)
%o A259477 (PARI) T(n,k)=max(if(k,n+1-k,n)*binomial(n+1,k+1),1)
%o A259477 for(n=0,10,for(k=0,n,print1(T(n,k)", "))) \\ _Charles R Greathouse IV_, Jun 29 2015
%K A259477 easy,nonn,tabl
%O A259477 0,2
%A A259477 _Vincent J. Matsko_, Jun 27 2015