A114117 Inverse of 1's counting matrix A114116.

1, 0, 1, -2, 1, 1, -1, -1, 1, 1, 0, -2, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, 0, -2, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, 0, -2, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 0

Author

Paul Barry, Nov 13 2005

Keywords

• easy
• sign
• tabl

Comments

Row sums are (1,1,0,0,0,.....) with g.f. 1+x. Diagonal sums have g.f. (1-x^2-x^3)/(1-x^3). Product of A114115 and the first difference matrix (1-x,x).

Examples

			Triangle begins
  1;
  0, 1;
 -2, 1, 1;
 -1,-1, 1, 1;
  0,-2, 0, 1, 1;
  0,-1,-1, 0, 1, 1;
  0, 0,-2, 0, 0, 1, 1;
  0, 0,-1,-1, 0, 0, 1, 1;
		

Crossrefs

Cf. A114115, A114116.

Formula

T(n, k) = Sum_{j=0..n} Sum_{i=0..n} C(floor((n+i)/2), j)*C(j, floor((n+i)/2))*(2*C(0, j-k)-C(1, j-k)).