A259569 Triangle T(n,k) read by rows, where T(n,k) is the number of k-dimensional faces of the polytope that is the convex hull of all permutations of the list (0,1,...,1,2), where there are n - 1 ones, for n > 0. T(0,0) is 1.

1, 2, 1, 6, 6, 1, 12, 24, 14, 1, 20, 60, 70, 30, 1, 30, 120, 210, 180, 62, 1, 42, 210, 490, 630, 434, 126, 1, 56, 336, 980, 1680, 1736, 1008, 254, 1, 72, 504, 1764, 3780, 5208, 4536, 2286, 510, 1, 90, 720, 2940, 7560, 13020, 15120, 11430, 5100, 1022, 1, 110, 990, 4620, 13860, 28644, 41580, 41910, 28050, 11242, 2046, 1

Offset: 0

Author

Vincent J. Matsko, Jun 30 2015

Keywords

• nonn
• tabl

Comments

It appears that these integers, with sign changes, are also in A138106.

Examples

			Triangle begins:
   1;
   2,  1;
   6,  6,  1;
  12, 24, 14,  1;
  20, 60, 70, 30,  1;
  ...
Row 2 describes a regular hexagon.
Row 3 describes the cuboctahedron.
		

Crossrefs

Row sums give A101052(n+1).

Cf. A138106.

Programs

Maple

T:= (n, k)-> `if`(n=k, 1, binomial(n+1, k+2)*(4*2^k-2)):
seq(seq(T(n,k), k=0..n), n=0..10);

Mathematica

Join @@ (CoefficientList[#,
     x] & /@ (Expand[
       D[((1 + 2 x) Exp[z (1 + 2 x)] - 2 (1 + x) Exp[z (1 + x)] + Exp[z] +
            x^2 Exp[z x])/x^2, {z, #}] /. z -> 0] & /@ Range[0, 10]))

Formula

T(n,n) = 1, n >= 0.

T(n,n-1) = 2^(n+1)-2, n > 0.

T(n,0) = n(n+1), n > 0.

T(n,k) = (n+1)*T(n-1,k)/(n-k-1), 0 <= k < n-1, n >= 2.

E.g.f.: ((2*x+1)*exp(z*(2*x+1)) - 2*(x+1)*exp(z*(x+1)) + x^2*exp(z*x)+exp(z))/x^2

Conjecture: Sum_{k=0..n-1} T(n,k)*x^(n-k-1) = x^(n+1) - 2(x+1)^(n+1) + (x+2)^(n+1). - Kevin J. Gomez, Jul 25 2017

T(n,n) = 1; T(n,k) = binomial(n+1,k+2)*(4*2^k - 2) for 0 <= k < n. - Aadesh Tikhe, May 25 2024