A111941 Matrix log of triangle A111940, which shifts columns left and up under matrix inverse; these terms are the result of multiplying each element in row n and column k by (n-k)!.

0, 1, 0, -1, -1, 0, 1, 1, 1, 0, -2, -1, -1, -1, 0, 4, 2, 1, 1, 1, 0, -12, -4, -2, -1, -1, -1, 0, 36, 12, 4, 2, 1, 1, 1, 0, -144, -36, -12, -4, -2, -1, -1, -1, 0, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0, 14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0, -86400, -14400, -2880, -576, -144, -36

Offset

0,11

Links

Table of n, a(n) for n=0..83.

Formula

T(n, k) = (-1)^k*T(n-k, 0) = (-1)^k*A111942(n-k) for n>=k>=0.

Example

Triangle begins:
0;
1, 0;
-1, -1, 0;
1, 1, 1, 0;
-2, -1, -1, -1, 0;
4, 2, 1, 1, 1, 0;
-12, -4, -2, -1, -1, -1, 0;
36, 12, 4, 2, 1, 1, 1, 0;
-144, -36, -12, -4, -2, -1, -1, -1, 0;
576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0;
14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-86400, -14400, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0;
518400, 86400, 14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-3628800, -518400, -86400, -14400, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0; ...
where, apart from signs, the columns are all the same (A111942).
...
Triangle A111940 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 the matrix inverse shifts columns left and up one place.
...
The matrix log of A111940, with factorial denominators, begins:
0;
1/1!, 0;
-1/2!, -1/1!, 0;
1/3!, 1/2!, 1/1!, 0;
-2/4!, -1/3!, -1/2!, -1/1!, 0;
4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-2880/10!, -576/9!, -144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
14400/11!, 2880/10!, 576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0; ...
Note that the square of the matrix log of A111940 begins:
0;
0, 0;
-1, 0, 0;
0, -1, 0, 0;
-1/12, 0, -1, 0, 0;
0, -1/12, 0, -1, 0, 0;
-1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/16632, 0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0; ...
where nonzero terms are negative unit fractions with denominators given by A002544:
[1, 12, 90, 560, 3150, 16632, 84084, 411840, ...,  C(2*n+1,n)*(n+1)^2, ...].

Prog

(PARI) {T(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); B=sum(i=1, #A, -(A^0-A)^i/i); return((n-k)!*B[n+1, k+1]))}
for(n=0, 16, for(k=0, n, print1(T(n, k, -1), ", ")); print(""))

Crossrefs

Cf. A111940 (triangle), A111942 (column 0), A110504 (variant).

Sequence in context: A308778 A127284 A120691 * A153462 A126310 A109086

Adjacent sequences: A111938 A111939 A111940 * A111942 A111943 A111944

Keyword

frac,sign,tabl

Author

Paul D. Hanna, Aug 23 2005

Status

approved