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A259569 Triangle of numbers where T(n,k) is the number of k-dimensional faces on 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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..65.

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

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

EXAMPLE

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.

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]))

CROSSREFS

Sequence in context: A229565 A259477 A208919 * A046651 A063007 A202190

Adjacent sequences:  A259566 A259567 A259568 * A259570 A259571 A259572

KEYWORD

nonn,tabl

AUTHOR

Vincent J. Matsko, Jun 30 2015

STATUS

approved

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Last modified February 16 12:48 EST 2019. Contains 320163 sequences. (Running on oeis4.)