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.

-1, 1, -1, 3, -1, -1, 9, -3, -1, -1, 27, -9, -3, -1, -1, 81, -27, -9, -3, -1, -1, 243, -81, -27, -9, -3, -1, -1, 729, -243, -81, -27, -9, -3, -1, -1, 2187, -729, -243, -81, -27, -9, -3, -1, -1, 6561, -2187, -729, -243, -81, -27, -9, -3, -1, -1, 19683, -6561, -2187

Offset: 0

Author

Roger L. Bagula, Dec 08 2008

Keywords

• sign
• easy
• tabl

Examples

			Triangle begins:
     -1;
      1,    -1;
      3,    -1,    -1;
      9,    -3,    -1,   -1;
     27,    -9,    -3,   -1,   -1;
     81,   -27,    -9,   -3,   -1,  -1;
    243,   -81,   -27,   -9,   -3,  -1,  -1;
    729,  -243,   -81,  -27,   -9,  -3,  -1, -1;
   2187,  -729,  -243,  -81,  -27,  -9,  -3, -1, -1;
   6561, -2187,  -729, -243,  -81, -27,  -9, -3, -1, -1;
  19683, -6561, -2187, -729, -243, -81, -27, -9, -3, -1, -1;
    ...
		

Crossrefs

Row sums (except row 0): A003462.

Cf. A057728, A152568, A152571, A152572.

Programs

Mathematica

b[0] = {-1}; b[1] = {1, -1};
b[n_] := b[n] = Join[{3^(n - 1)}, {-b[n - 1][[1]]}, Table[b[n - 1][[i]], {i, 2, Length[b[n - 1]]}]];
Flatten[Table[b[n], {n, 0, 10}]]

Maxima

T(n,k) := if k = n then -1 else if k = 0 then 3^(n - 1) else -3^(n - k - 1)$
create_list(T(n, k), n, 0, 20, k, 0, n); /* Franck Maminirina Ramaharo, Jan 08 2019 */

Formula

From Franck Maminirina Ramaharo, Jan 08 2019: (Start)

G.f.: -(1 - 4*y + 2*x*y^2)/(1 - (3 + x)*y + 3*x*y^2).

E.g.f.: -(6 - 2*x - (3 - 2*x)*exp(3*y) + (6 - 3*x)*exp(x*y))/(9 - 3*x). (End)

Extensions

Edited by Franck Maminirina Ramaharo, Jan 08 2019