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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.
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
OFFSET
0,2
COMMENTS
It appears that these integers, with sign changes, are also in A138106.
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
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.
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]))
CROSSREFS
Row sums give A101052(n+1).
Cf. A138106.
Sequence in context: A259477 A208919 A347580 * A046651 A063007 A331430
KEYWORD
nonn,tabl
AUTHOR
Vincent J. Matsko, Jun 30 2015
STATUS
approved