A111843 Matrix log of triangle A111840, which shifts columns left and up under matrix cube; these terms are the result of multiplying each element in row n and column k by (n-k)!.

0, 1, 0, 3, 3, 0, 27, 9, 9, 0, 486, 81, 27, 27, 0, 7776, 1458, 243, 81, 81, 0, -2423196, 23328, 4374, 729, 243, 243, 0, -97338996, -7269588, 69984, 13122, 2187, 729, 729, 0, 5883879500784, -292016988, -21808764, 209952, 39366, 6561, 2187, 2187, 0

Offset

0,4

Comments

Column k equals 3^k multiplied by column 0 (A111844) when ignoring zeros above the diagonal.

Links

Table of n, a(n) for n=0..44.

Formula

T(n, k) = 3^k*T(n-k, 0) = 3^k*A111844(n-k) for n>=k>=0.

Example

Matrix log of A111840, with factorial denominators, begins:
0;
1/1!, 0;
3/2!, 3/1!, 0;
27/3!, 9/2!, 9/1!, 0;
486/4!, 81/3!, 27/2!, 27/1!, 0;
7776/5!, 1458/4!, 243/3!, 81/2!, 81/1!, 0;
-2423196/6!, 23328/5!, 4374/4!, 729/3!, 243/2!, 243/1!, 0;

Prog

(PARI) T(n, k, q=3)=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]))

Crossrefs

Cf. A111840 (triangle), A111844 (column 0), A111815 (variant), A111941 (q=-1), A111810 (q=2), A111848 (q=4).

Sequence in context: A137259 A166553 A285863 * A119537 A338148 A338144

Adjacent sequences: A111840 A111841 A111842 * A111844 A111845 A111846

Keyword

frac,sign,tabl

Author

Paul D. Hanna, Aug 23 2005

Status

approved