A110503 Triangle, read by rows, which shifts one column left under matrix inverse.

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

Offset: 0

Author

Paul D. Hanna, Jul 23 2005

Keywords

• sign
• tabl

Comments

The unsigned columns of the matrix logarithm of this triangle are all equal to A110504.

Examples

			Triangle begins:
  1;
  1,  1;
  1, -1,  1;
  1, -2,  1,  1;
  1, -1,  1, -1,  1;
  1, -1,  1, -2,  1,  1;
  1, -1,  1, -1,  1, -1,  1;
  1, -1,  1, -1,  1, -2,  1,  1;
  1, -1,  1, -1,  1, -1,  1, -1,  1;
  1, -1,  1, -1,  1, -1,  1, -2,  1,  1; ...
The matrix inverse drops the first column:
   1;
  -1,  1;
  -2,  1,  1;
  -1,  1, -1,  1;
  -1,  1, -2,  1,  1;
  -1,  1, -1,  1, -1,  1; ...
The matrix logarithm equals:
     0;
    1/1!,     0;
    3/2!,   -1/1!,   0;
    7/3!,   -3/2!,  1/1!,   0;
   30/4!,   -7/3!,  3/2!, -1/1!,  0;
  144/5!,  -30/4!,  7/3!, -3/2!, 1/1!,   0;
  876/6!, -144/5!, 30/4!, -7/3!, 3/2!, -1/1!, 0; ...
unsigned columns of which all equal A110505.
		

Crossrefs

Cf. A110504 (matrix log), A110505 (column 0 of log).

Cf. A111940 (variant).

Programs

PARI

T(n,k)=matrix(n+1,n+1,r,c,if(r>=c, if(r==c || c%2==1,1,if(r%2==0 && r==c+2,-2,-1))))[n+1,k+1]

Formula

T(n, k) = +1 when k == 0 (mod 2), T(n, k)=-1 when k == 1 (mod 2), except for T(k+2, k) = -2 when k == 1 (mod 2) and T(n, n) = 1.

G.f. for column k of matrix power A110503^m (ignoring leading zeros): cos(m*arccos(1-x^2/2)) + (-1)^k*sin(m*arccos(1-x^2/2))*(1-x/2)/sqrt(1-x^2/4)*(1+x)/(1-x).