A051632 Rows of triangle formed using Pascal's rule except we begin and end the n-th row with n-2.

-2, -1, -1, 0, -2, 0, 1, -2, -2, 1, 2, -1, -4, -1, 2, 3, 1, -5, -5, 1, 3, 4, 4, -4, -10, -4, 4, 4, 5, 8, 0, -14, -14, 0, 8, 5, 6, 13, 8, -14, -28, -14, 8, 13, 6, 7, 19, 21, -6, -42, -42, -6, 21, 19, 7, 8, 26, 40, 15, -48, -84, -48, 15, 40, 26, 8, 9, 34, 66, 55, -33, -132, -132

Offset: 0

Author

Asher Auel

Keywords

• easy
• nice
• sign
• tabl

Comments

Row sums are -2.

Examples

			Contribution from Roger L. Bagula, Feb 17 2009: (Start)
The rows of the triangle, negated, are:
{2},
{1, 1},
{0, 2, 0},
{-1, 2, 2, -1},
{-2, 1, 4, 1, -2},
{-3, -1, 5,5, -1, -3},
{-4, -4, 4, 10, 4, -4, -4},
{-5, -8, 0, 14, 14, 0, -8, -5},
{-6, -13, -8, 14, 28, 14, -8, -13, -6},
{-7, -19, -21, 6, 42,42, 6, -21, -19, -7},
{-8, -26, -40, -15, 48, 84, 48, -15, -40, -26, -8} (End)
		

Links

• Reinhard Zumkeller, Rows n=0..100 of triangle, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

A156644 [From Roger L. Bagula, Feb 17 2009]

Cf. A007318.

Programs

Haskell

a051632 n k = a051632_tabl !! n !! k
a051632_list = concat a051632_tabl
a051632_tabl = iterate (\rs -> zipWith (+) ([1] ++ rs) (rs ++ [1])) [-2]
-- Reinhard Zumkeller, Aug 21 2011

Mathematica

t[n_, k_] = ((2*k + 1 - n)/(k + 1))*Binomial[n, k] + ((1 - n + 2 (-k + n))/(1 - k + n)) Binomial[n, -k + n];
Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}];
Flatten[%] (* Roger L. Bagula , Feb 17 2009 *)

Formula

-t(n,k)=((2*k + 1 - n)/(k + 1))*Binomial[n, k] + ((1 - n + 2 (-k + n))/(1 - k + n)) Binomial[n, -k + n]. [From Roger L. Bagula, Feb 17 2009]