A080233 Triangle T(n,k) obtained by taking differences of consecutive pairs of row elements of Pascal's triangle A007318.

1, 1, 0, 1, 1, -1, 1, 2, 0, -2, 1, 3, 2, -2, -3, 1, 4, 5, 0, -5, -4, 1, 5, 9, 5, -5, -9, -5, 1, 6, 14, 14, 0, -14, -14, -6, 1, 7, 20, 28, 14, -14, -28, -20, -7, 1, 8, 27, 48, 42, 0, -42, -48, -27, -8, 1, 9, 35, 75, 90, 42, -42, -90, -75, -35, -9

Offset: 0

Author

Paul Barry, Feb 10 2003

Keywords

• easy
• sign
• tabl

Comments

Row sums are 1,1,1,1,1,1 with g.f. 1/(1-x). Can also be obtained from triangle A080232 by taking sums of pairs of consecutive row elements.

Mirror image of triangle in A156644. - Philippe Deléham, Feb 14 2009

Examples

			Triangle begins as:
  1;
  1, 0;
  1, 1, -1;
  1, 2,  0, -2;
  1, 3,  2, -2, -3;
  1, 4,  5,  0, -5,  -4;
  1, 5,  9,  5, -5,  -9,  -5;
  1, 6, 14, 14,  0, -14, -14,  -6;
  1, 7, 20, 28, 14, -14, -28, -20,  -7;
  1, 8, 27, 48, 42,   0, -42, -48, -27,  -8;
  1, 9, 35, 75, 90,  42, -42, -90, -75, -35, -9;
  ...
		

Crossrefs

Row sums give A000012.

Cf. A007318, A080232.

Programs

Mathematica

Table[Binomial[n, k] - Binomial[n, k - 1], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

PARI

{T(n, k) = if( n<0 || k>n, 0, binomial(n, k) - binomial(n, k-1))}; /* Michael Somos, Nov 25 2016 */

Formula

T(n, k) = if(k>n, 0, binomial(n, k)-binomial(n, k-1)).