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Triangle T(n,k) read by rows: T(n,n) = -1, T(n,0) = 2^(n - 1), T(n,k) = -2^(n - k - 1), 1 <= k <= n - 1.
6

%I #11 Jan 10 2019 02:19:16

%S -1,1,-1,2,-1,-1,4,-2,-1,-1,8,-4,-2,-1,-1,16,-8,-4,-2,-1,-1,32,-16,-8,

%T -4,-2,-1,-1,64,-32,-16,-8,-4,-2,-1,-1,128,-64,-32,-16,-8,-4,-2,-1,-1,

%U 256,-128,-64,-32,-16,-8,-4,-2,-1,-1,512,-256,-128,-64,-32,-16,-8,-4,-2

%N Triangle T(n,k) read by rows: T(n,n) = -1, T(n,0) = 2^(n - 1), T(n,k) = -2^(n - k - 1), 1 <= k <= n - 1.

%C Except for n = 0, the row sums are zero.

%F From _Franck Maminirina Ramaharo_, Jan 08 2019: (Start)

%F G.f.: -(1 - 3*y + 2*x*y^2)/(1 - (2 + x)*y + 2*x*y^2).

%F E.g.f.: (exp(2*y) - exp(x*y))*(1 - x)/(2 - x) - 1. (End)

%e Triangle begins:

%e -1;

%e 1, -1;

%e 2, -1, -1;

%e 4, -2, -1, -1;

%e 8, -4, -2, -1, -1;

%e 16, -8, -4, -2, -1, -1;

%e 32, -16, -8, -4, -2, -1, -1;

%e 64, -32, -16, -8, -4, -2, -1, -1;

%e 128, -64, -32, -16, -8, -4, -2, -1, -1;

%e 256, -128, -64, -32, -16, -8, -4, -2, -1, -1;

%e 512, -256, -128, -64, -32, -16, -8, -4, -2, -1, -1;

%e ...

%t b[0] = {-1}; b[1] = {1, -1};

%t b[n_] := b[n] = Join[{2^(n - 1)}, {-b[n - 1][[1]]}, Table[b[n - 1][[i]], {i, 2, Length[b[n - 1]]}]]

%t Flatten[Table[b[n], {n, 0, 10}]]

%o (Maxima)

%o T(n, k) := if k = n then -1 else if k = 0 then 2^(n - 1) else -2^(n - k - 1)$

%o create_list(T(n, k), n, 0, 20, k, 0, n); /* _Franck Maminirina Ramaharo_, Jan 08 2019 */

%Y Cf. A057728, A152570, A152571, A152572.

%K sign,tabl,easy

%O 0,4

%A _Roger L. Bagula_, Dec 08 2008

%E Unrelated material removed by the Assoc. Eds. of the OEIS, Jun 07 2010