A110504 Triangle, read by rows, which equals the matrix logarithm of the triangle A110503.

0, 1, 0, 3, -1, 0, 7, -3, 1, 0, 30, -7, 3, -1, 0, 144, -30, 7, -3, 1, 0, 876, -144, 30, -7, 3, -1, 0, 6084, -876, 144, -30, 7, -3, 1, 0, 48816, -6084, 876, -144, 30, -7, 3, -1, 0, 438624, -48816, 6084, -876, 144, -30, 7, -3, 1, 0, 4389120, -438624, 48816, -6084, 876, -144, 30, -7, 3, -1, 0

Offset

0,4

Comments

The unsigned columns this triangle are all equal to A110505. Triangle A110503 shifts one column left under matrix inverse.

Links

Table of n, a(n) for n=0..65.

Formula

T(n, k) = (-1)^k*A110505(n-k).

Example

Triangle begins:
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;
6084/7!, -876/6!, 144/5!, -30/4!, 7/3!, -3/2!, 1/1!, 0; ...
Unsigned columns all equal A110505.
Exponential function of matrix equals A110503:
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; ...

Prog

(PARI) T(n, k)=local(M=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))))); sum(i=1, #M, -(M^0-M)^i/i)[n+1, k+1]

Crossrefs

Cf. A110503 (matrix exponential), A110505 (unsigned columns).

Sequence in context: A130888 A010601 A176108 * A361475 A111246 A206306

Adjacent sequences: A110501 A110502 A110503 * A110505 A110506 A110507

Keyword

sign,tabl

Author

Paul D. Hanna, Jul 23 2005

Status

approved