A152572 Triangle T(n,k) read by rows: T(n,n) = -1, T(n,0) = 5^(n - 1), T(n,k) = -5^(n - k - 1), 1 <= k <= n - 1.

-1, 1, -1, 5, -1, -1, 25, -5, -1, -1, 125, -25, -5, -1, -1, 625, -125, -25, -5, -1, -1, 3125, -625, -125, -25, -5, -1, -1, 15625, -3125, -625, -125, -25, -5, -1, -1, 78125, -15625, -3125, -625, -125, -25, -5, -1, -1, 390625, -78125, -15625, -3125, -625, -125, -25, -5, -1, -1

Offset: 0

Author

Roger L. Bagula, Dec 08 2008

Keywords

• sign
• easy
• tabl

Examples

			Triangle begins:
       -1;
        1,      -1;
        5,      -1,     -1;
       25,      -5,     -1,     -1;
      125,     -25,     -5,     -1,    -1;
      625,    -125,    -25,     -5,    -1,   -1;
     3125,    -625,   -125,    -25,    -5,   -1,   -1;
    15625,   -3125,   -625,   -125,   -25,   -5,   -1,  -1;
    78125,  -15625,  -3125,   -625,  -125,  -25,   -5,  -1, -1;
   390625,  -78125, -15625,  -3125,  -625, -125,  -25,  -5, -1, -1;
  1953125, -390625, -78125, -15625, -3125, -625, -125, -25, -5, -1, -1;
      ...
		

Crossrefs

Row sums (except row 0): A125833.

Cf. A057728, A152568, A152570, A152571.

Programs

Mathematica

b[0] = {-1}; b[1] = {1, -1};
b[n_] := b[n] = Join[{5^(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 5^(n - 1) else -5^(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 - 6*y + 2*x*y^2)/(1 - (5 + x)*y + 5*x*y^2).

E.g.f.: -(10 - 2*x - (5 - 2*x)*exp(5*y) + (20 - 5*x)*exp(x*y))/(25 - 5*x). (End)

Extensions

Edited by Franck Maminirina Ramaharo, Jan 08 2019