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.

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, 63, 14, 1, 56, 196, 336, 350, 224, 84, 16, 1, 72, 288, 588, 756, 630, 336, 108, 18, 1, 90, 405, 960, 1470, 1512, 1050, 480, 135, 20, 1, 110, 550, 1485, 2640, 3234, 2772, 1650, 660, 165, 22, 1

Offset

0,2

Links

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Formula

T(n,0) = n*(n+1), n > 0; T(n,k) = (n+1-k)*binomial(n+1,k+1), 1 <= k <= n.

E.g.f.: ((x+1)*(z+1)+1)*exp(z)*(exp(x*z)-1)/x + 1.

From Vincent J. Matsko, Jul 18 2015: (Start)

O.g.f.: (1/(1-(x+1)*y)^2-(x+1)/(1-y)^2)/x + 1/((1-(x+1)y)*(1-y))+1.

G.f. for rows (n > 0): (((x+1)^n-1)*(x+n+2))/x-n. (End)

Example

Triangle begins:
1;
2, 1;
6, 6, 1;
12, 18, 8, 1;
20, 40, 30, 10, 1;
...

Mathematica

Join @@ (CoefficientList[#,
     x] & /@ (Expand[
       D[((x + 1) (z + 1) + 1) Exp[z] (Exp[x z] - 1)/x + 1, {z, #}] /.
        z -> 0] & /@ Range[0, 10])) (* Vincent J. Matsko, Jun 30 2015 *)
Flatten[Table[
  CoefficientList[
   D[(1/(1 - (x + 1)*y)^2 - (x + 1)/(1 - y)^2)/x +
       1/((1 - (x + 1) y)*(1 - y)) + 1, {y, k}]/Factorial[k] /. y -> 0,
   x], {k, 0, 10}]] (* Vincent J. Matsko, Jul 18 2015 *)

Prog

(PARI) T(n, k)=max(if(k, n+1-k, n)*binomial(n+1, k+1), 1)
for(n=0, 10, for(k=0, n, print1(T(n, k)", "))) \\ Charles R Greathouse IV, Jun 29 2015

Crossrefs

Sequence in context: A133314 A208909 A229565 * A208919 A347580 A259569

Adjacent sequences: A259474 A259475 A259476 * A259478 A259479 A259480

Keyword

easy,nonn,tabl

Author

Vincent J. Matsko, Jun 27 2015

Status

approved