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A152570 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.

%I A152570 #11 Jan 10 2019 02:19:53
%S A152570 -1,1,-1,3,-1,-1,9,-3,-1,-1,27,-9,-3,-1,-1,81,-27,-9,-3,-1,-1,243,-81,
%T A152570 -27,-9,-3,-1,-1,729,-243,-81,-27,-9,-3,-1,-1,2187,-729,-243,-81,-27,
%U A152570 -9,-3,-1,-1,6561,-2187,-729,-243,-81,-27,-9,-3,-1,-1,19683,-6561,-2187
%N A152570 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 A152570 From _Franck Maminirina Ramaharo_, Jan 08 2019: (Start)
%F A152570 G.f.: -(1 - 4*y + 2*x*y^2)/(1 - (3 + x)*y + 3*x*y^2).
%F A152570 E.g.f.: -(6 - 2*x - (3 - 2*x)*exp(3*y) + (6 - 3*x)*exp(x*y))/(9 - 3*x). (End)
%e A152570 Triangle begins:
%e A152570      -1;
%e A152570       1,    -1;
%e A152570       3,    -1,    -1;
%e A152570       9,    -3,    -1,   -1;
%e A152570      27,    -9,    -3,   -1,   -1;
%e A152570      81,   -27,    -9,   -3,   -1,  -1;
%e A152570     243,   -81,   -27,   -9,   -3,  -1,  -1;
%e A152570     729,  -243,   -81,  -27,   -9,  -3,  -1, -1;
%e A152570    2187,  -729,  -243,  -81,  -27,  -9,  -3, -1, -1;
%e A152570    6561, -2187,  -729, -243,  -81, -27,  -9, -3, -1, -1;
%e A152570   19683, -6561, -2187, -729, -243, -81, -27, -9, -3, -1, -1;
%e A152570     ...
%t A152570 b[0] = {-1}; b[1] = {1, -1};
%t A152570 b[n_] := b[n] = Join[{3^(n - 1)}, {-b[n - 1][[1]]}, Table[b[n - 1][[i]], {i, 2, Length[b[n - 1]]}]];
%t A152570 Flatten[Table[b[n], {n, 0, 10}]]
%o A152570 (Maxima)
%o A152570 T(n,k) := if k = n then -1 else if k = 0 then 3^(n - 1) else -3^(n - k - 1)$
%o A152570 create_list(T(n, k), n, 0, 20, k, 0, n); /* _Franck Maminirina Ramaharo_, Jan 08 2019 */
%Y A152570 Row sums (except row 0): A003462.
%Y A152570 Cf. A057728, A152568, A152571, A152572.
%K A152570 sign,easy,tabl
%O A152570 0,4
%A A152570 _Roger L. Bagula_, Dec 08 2008
%E A152570 Edited by _Franck Maminirina Ramaharo_, Jan 08 2019