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%I #11 Jan 10 2019 02:19:53
%S -1,1,-1,3,-1,-1,9,-3,-1,-1,27,-9,-3,-1,-1,81,-27,-9,-3,-1,-1,243,-81,
%T -27,-9,-3,-1,-1,729,-243,-81,-27,-9,-3,-1,-1,2187,-729,-243,-81,-27,
%U -9,-3,-1,-1,6561,-2187,-729,-243,-81,-27,-9,-3,-1,-1,19683,-6561,-2187
%N Triangle T(n,k) read by rows: T(n,n) = -1, T(n,0) = 3^(n - 1), T(n,k) = -3^(n - k - 1), 1 <= k <= n - 1.
%F From _Franck Maminirina Ramaharo_, Jan 08 2019: (Start)
%F G.f.: -(1 - 4*y + 2*x*y^2)/(1 - (3 + x)*y + 3*x*y^2).
%F E.g.f.: -(6 - 2*x - (3 - 2*x)*exp(3*y) + (6 - 3*x)*exp(x*y))/(9 - 3*x). (End)
%e Triangle begins:
%e -1;
%e 1, -1;
%e 3, -1, -1;
%e 9, -3, -1, -1;
%e 27, -9, -3, -1, -1;
%e 81, -27, -9, -3, -1, -1;
%e 243, -81, -27, -9, -3, -1, -1;
%e 729, -243, -81, -27, -9, -3, -1, -1;
%e 2187, -729, -243, -81, -27, -9, -3, -1, -1;
%e 6561, -2187, -729, -243, -81, -27, -9, -3, -1, -1;
%e 19683, -6561, -2187, -729, -243, -81, -27, -9, -3, -1, -1;
%e ...
%t b[0] = {-1}; b[1] = {1, -1};
%t b[n_] := b[n] = Join[{3^(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 3^(n - 1) else -3^(n - k - 1)$
%o create_list(T(n, k), n, 0, 20, k, 0, n); /* _Franck Maminirina Ramaharo_, Jan 08 2019 */
%Y Row sums (except row 0): A003462.
%Y Cf. A057728, A152568, A152571, A152572.
%K sign,easy,tabl
%O 0,4
%A _Roger L. Bagula_, Dec 08 2008
%E Edited by _Franck Maminirina Ramaharo_, Jan 08 2019
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