A152571 Triangle T(n,k) read by rows: T(n,n) = -1, T(n,0) = 4^(n - 1), T(n,k) = -4^(n - k - 1), 1 <= k <= n - 1.

-1, 1, -1, 4, -1, -1, 16, -4, -1, -1, 64, -16, -4, -1, -1, 256, -64, -16, -4, -1, -1, 1024, -256, -64, -16, -4, -1, -1, 4096, -1024, -256, -64, -16, -4, -1, -1, 16384, -4096, -1024, -256, -64, -16, -4, -1, -1, 65536, -16384, -4096, -1024, -256, -64, -16, -4, -1, -1

Offset: 0

Author

Roger L. Bagula, Dec 08 2008

Keywords

• sign
• easy
• tabl

Examples

			Triangle begins:
      -1;
       1,     -1;
       4,     -1,     -1;
      16,     -4,     -1,    -1;
      64,    -16,     -4,    -1,    -1;
     256,    -64,    -16,    -4,    -1,   -1;
    1024,   -256,    -64,   -16,    -4,   -1,  -1;
    4096,  -1024,   -256,   -64,   -16,   -4,  -1,  -1;
   16384,  -4096,  -1024,  -256,   -64,  -16,  -4,  -1, -1;
   65536, -16384,  -4096, -1024,  -256,  -64, -16,  -4, -1, -1;
  262144, -65536, -16384, -4096, -1024, -256, -64, -16, -4, -1, -1;
     ...
		

Crossrefs

Row sums (except row 0): A020988.

Cf. A057728, A152568, A152570, A152572.

Programs

Mathematica

b[0] = {-1}; b[1] = {1, -1};
b[n_] := b[n] = Join[{4^(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 4^(n - 1) else -4^(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 - 5*y + 2*x*y^2)/(1 - (4 + x)*y + 4*x*y^2).

E.g.f.: -(4 - x - (2 - x)*exp(4*y) + (6 - 2*x)*exp(x*y))/(8 - 2*x). (End)

Extensions

Edited by Franck Maminirina Ramaharo, Jan 08 2019