A111828 Matrix log of triangle A111825, which shifts columns left and up under matrix 6th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

0, 1, 0, -4, 6, 0, 42, -24, 36, 0, 7296, 252, -144, 216, 0, -7931976, 43776, 1512, -864, 1296, 0, -45557382240, -47591856, 262656, 9072, -5184, 7776, 0, 3064554175021200, -273344293440, -285551136, 1575936, 54432, -31104, 46656, 0, 801993619807364206080

Offset

0,4

Comments

Column k equals 6^k multiplied by column 0 (A111829) when ignoring zeros above the diagonal.

Links

Table of n, a(n) for n=0..36.

Formula

T(n, k) = 6^k*T(n-k, 0) = A111829(n-k) for n>=k>=0.

Example

Matrix log of A111825, with factorial denominators, begins:
0;
1/1!, 0;
-4/2!, 6/1!, 0;
42/3!, -24/2!, 36/1!, 0;
7296/4!, 252/3!, -144/2!, 216/1!, 0;
-7931976/5!, 43776/4!, 1512/3!, -864/2!, 1296/1!, 0; ...

Prog

(PARI) T(n, k, q=6)=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 || j==1, B[i, j]=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. A111825, A111829 (column 0); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111818 (q=4), A111823 (q=5), A111833 (q=7), A111838 (q=8).

Sequence in context: A288422 A268437 A265685 * A013249 A013253 A013246

Adjacent sequences: A111825 A111826 A111827 * A111829 A111830 A111831

Keyword

frac,sign,tabl

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Status

approved