A152568 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.

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

Offset: 0

Author

Roger L. Bagula, Dec 08 2008

Keywords

• sign
• tabl
• easy

Comments

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

Examples

			Triangle begins:
   -1;
    1,   -1;
    2,   -1,   -1;
    4,   -2,   -1,  -1;
    8,   -4,   -2,  -1,  -1;
   16,   -8,   -4,  -2,  -1,  -1;
   32,  -16,   -8,  -4,  -2,  -1, -1;
   64,  -32,  -16,  -8,  -4,  -2, -1, -1;
  128,  -64,  -32, -16,  -8,  -4, -2, -1, -1;
  256, -128,  -64, -32, -16,  -8, -4, -2, -1, -1;
  512, -256, -128, -64, -32, -16, -8, -4, -2, -1, -1;
  ...
		

Crossrefs

Cf. A057728, A152570, A152571, A152572.

Programs

Mathematica

b[0] = {-1}; b[1] = {1, -1};
b[n_] := b[n] = Join[{2^(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 2^(n - 1) else -2^(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 - 3*y + 2*x*y^2)/(1 - (2 + x)*y + 2*x*y^2).

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

Extensions

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