A111940 Triangle P, read by rows, that satisfies [P^-1](n,k) = P(n+1,k+1) for n >= k >= 0, with P(k,k)=1 and P(k+1,1)=P(k+1,0) for k >= 0, where [P^-1] denotes the matrix inverse of P.

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

Offset: 0

Author

Paul D. Hanna, Aug 23 2005

Keywords

• sign
• tabl

Examples

			Triangle P begins:
   1;
   1,  1;
  -1, -1,  1;
   0,  0,  1,  1;
   0,  0, -1, -1,  1;
   0,  0,  0,  0,  1,  1;
   0,  0,  0,  0, -1, -1,  1;
   0,  0,  0,  0,  0,  0,  1,  1;
   0,  0,  0,  0,  0,  0, -1, -1,  1; ...
where P^-1 shifts columns left and up one place:
   1;
  -1,  1;
   0,  1,  1;
   0, -1, -1,  1;
   0,  0,  0,  1,  1;
   0,  0,  0, -1, -1,  1; ...
		

Crossrefs

Cf. A111941 (matrix log), A111942, A110503 (variant).

Programs

PARI

{P(n,k,q=-1) = local(A=Mat(1),B); if(n
				

Formula

The g.f. of column k of matrix power P^m (ignoring leading zeros) is:

cos(m*arccos(1-x^2/2)) + (-1)^k * sin(m*arccos(1-x^2/2)) * (1-x/2) / sqrt(1-x^2/4).