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

0,1

Links

Table of n, a(n) for n=0..90.

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).

Example

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; ...

Prog

(PARI) {P(n, k, q=-1) = local(A=Mat(1), B); if(n<k||k<0, 0, for(m=1, n+1, B = matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, if(j==1, B[i, j]=(A^q)[i-1, 1], B[i, j] = (A^q)[i-1, j-1])); )); A=B); return(A[n+1, k+1]))}
for(n=0, 16, for(k=0, n, print1(P(n, k, -1), ", ")); print(""))

Crossrefs

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

Sequence in context: A175608 A285467 A194679 * A129572 A070950 A071031

Adjacent sequences: A111937 A111938 A111939 * A111941 A111942 A111943

Keyword

sign,tabl

Author

Paul D. Hanna, Aug 23 2005

Status

approved