OFFSET
0,4
FORMULA
T^(m+3) = SHIFT_UP(T^(m+1) - T^m) - D*T^m for all m where diagonal matrix D = [0, 1, 2, 3, ...] and SHIFT_UP shifts each column up 1 row.
EXAMPLE
Triangle T begins:
1;
1,1;
3,2,1;
15,6,3,1;
96,36,9,4,1;
735,258,63,12,5,1;
6447,2190,492,96,15,6,1;
63120,20988,4545,804,135,18,7,1;
677739,222042,46935,7980,1200,180,21,8,1; ...
Matrix cube T^3 starts:
1;
3,1;
15,6,1;
96,36,9,1;
735,258,63,12,1;
6447,2190,492,96,15,1; ...
which equals SHIFT_UP(T) - D where
D is the diagonal matrix [0,1,2,3,...].
PROG
(PARI) {T(n, k)=local(M=matrix(n+3, n+3)); M=M^0; for(i=1, n, M=matrix(n+3, n+3, r, c, if(r>=c, if(r==c, 1, if(r==c+1, c, (M^3)[r-1, c]))))); return((M^3)[n+1, k+1])}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jun 24 2005
STATUS
approved