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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.

%I A259569 #51 Jun 22 2024 16:03:34
%S A259569 1,2,1,6,6,1,12,24,14,1,20,60,70,30,1,30,120,210,180,62,1,42,210,490,
%T A259569 630,434,126,1,56,336,980,1680,1736,1008,254,1,72,504,1764,3780,5208,
%U A259569 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
%N 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.
%C A259569 It appears that these integers, with sign changes, are also in A138106.
%F A259569 T(n,n) = 1, n >= 0.
%F A259569 T(n,n-1) = 2^(n+1)-2, n > 0.
%F A259569 T(n,0) = n(n+1), n > 0.
%F A259569 T(n,k) = (n+1)*T(n-1,k)/(n-k-1), 0 <= k < n-1, n >= 2.
%F A259569 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
%F A259569 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
%F A259569 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
%e A259569 Triangle begins:
%e A259569    1;
%e A259569    2,  1;
%e A259569    6,  6,  1;
%e A259569   12, 24, 14,  1;
%e A259569   20, 60, 70, 30,  1;
%e A259569   ...
%e A259569 Row 2 describes a regular hexagon.
%e A259569 Row 3 describes the cuboctahedron.
%p A259569 T:= (n, k)-> `if`(n=k, 1, binomial(n+1, k+2)*(4*2^k-2)):
%p A259569 seq(seq(T(n,k), k=0..n), n=0..10);
%t A259569 Join @@ (CoefficientList[#,
%t A259569      x] & /@ (Expand[
%t A259569        D[((1 + 2 x) Exp[z (1 + 2 x)] - 2 (1 + x) Exp[z (1 + x)] + Exp[z] +
%t A259569             x^2 Exp[z x])/x^2, {z, #}] /. z -> 0] & /@ Range[0, 10]))
%Y A259569 Row sums give A101052(n+1).
%Y A259569 Cf. A138106.
%K A259569 nonn,tabl
%O A259569 0,2
%A A259569 _Vincent J. Matsko_, Jun 30 2015