A228053 A triangle formed like Pascal's triangle, but with (-1)^(n+1) on the borders instead of 1.

-1, 1, 1, -1, 2, -1, 1, 1, 1, 1, -1, 2, 2, 2, -1, 1, 1, 4, 4, 1, 1, -1, 2, 5, 8, 5, 2, -1, 1, 1, 7, 13, 13, 7, 1, 1, -1, 2, 8, 20, 26, 20, 8, 2, -1, 1, 1, 10, 28, 46, 46, 28, 10, 1, 1, -1, 2, 11, 38, 74, 92, 74, 38, 11, 2, -1, 1, 1, 13, 49, 112, 166, 166, 112

Offset: 0

Author

T. D. Noe, Aug 07 2013

Keywords

• sign
• tabl

Comments

This sequence is almost the same as A026637.

T(n,k) = A026637(n-2,k-1) for n > 3, 1 < k < n-1. - Reinhard Zumkeller, Aug 08 2013

Examples

			Triangle begins:
  -1,
  1, 1,
  -1, 2, -1,
  1, 1, 1, 1,
  -1, 2, 2, 2, -1,
  1, 1, 4, 4, 1, 1,
  -1, 2, 5, 8, 5, 2, -1,
  1, 1, 7, 13, 13, 7, 1, 1,
  -1, 2, 8, 20, 26, 20, 8, 2, -1,
  1, 1, 10, 28, 46, 46, 28, 10, 1, 1,
  -1, 2, 11, 38, 74, 92, 74, 38, 11, 2, -1
		

Links

• T. D. Noe, Rows n = 0..50 of triangle, flattened

Crossrefs

Cf. A007318 (Pascal's triangle), A026637 (many terms in common).

Cf. A051601 (n on the borders), A137688 (2^n on borders).

Cf. A097073 (row sums).

Cf. A227074 (4^n edges), A227075 (3^n edges), A227076 (5^n edges).

Programs

Haskell

a228053 n k = a228053_tabl !! n !! k
a228053_row n = a228053_tabl !! n
a228053_tabl = iterate (\row@(i:_) -> zipWith (+)
   ([- i] ++ tail row ++ [0]) ([0] ++ init row ++ [- i])) [- 1]
  -- Reinhard Zumkeller, Aug 08 2013

Mathematica

t = {}; Do[r = {}; Do[If[k == 0 || k == n, m = (-1)^(n+1), m = t[[n, k]] + t[[n, k + 1]]]; r = AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t = Flatten[t]