OFFSET
0,4
FORMULA
Column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.
EXAMPLE
Triangle T begins:
1;
1, 1;
4, 2, 1;
28, 10, 3, 1;
268, 78, 18, 4, 1;
3164, 798, 156, 28, 5, 1;
43672, 9874, 1714, 268, 40, 6, 1;
682632, 141282, 22368, 3164, 420, 54, 7, 1;
11834536, 2273730, 333910, 43672, 5320, 618, 70, 8, 1;
224283416, 40400466, 5566728, 682632, 77720, 8378, 868, 88, 9, 1; ...
where column k of T = column 0 of T^(k+1)
with column 0 of T = column 0 of T^4 shift right:
column 1 of T = column 0 of T^2;
column 2 of T = column 0 of T^3;
column 3 of T = column 0 of T^4.
Matrix square of T, T^2, begins:
1;
2, 1;
10, 4, 1;
78, 26, 6, 1;
798, 232, 48, 8, 1;
9874, 2578, 486, 76, 10, 1;
141282, 33764, 5888, 864, 110, 12, 1;
2273730, 503910, 82210, 11396, 1390, 150, 14, 1; ...
where column k of T^2 = column 1 of T^(k+1):
column 0 of T^2 = column 1 of T;
column 2 of T^2 = column 1 of T^3;
column 3 of T^2 = column 1 of T^4.
Matrix cube of T, T^3, begins:
1;
3, 1;
18, 6, 1;
156, 48, 9, 1;
1714, 486, 90, 12, 1;
22368, 5888, 1050, 144, 15, 1;
333910, 82210, 14046, 1908, 210, 18, 1;
5566728, 1289928, 211182, 28072, 3120, 288, 21, 1; ...
where column k of T^3 = column 2 of T^(k+1):
column 0 of T^3 = column 2 of T;
column 1 of T^3 = column 2 of T^2;
column 3 of T^3 = column 2 of T^4.
Matrix 4th power of T, T^4, begins:
1;
4, 1;
28, 8, 1;
268, 76, 12, 1;
3164, 864, 144, 16, 1;
43672, 11396, 1908, 232, 20, 1;
682632, 170000, 28072, 3520, 340, 24, 1;
11834536, 2814832, 454848, 57408, 5820, 468, 28, 1; ...
where column k of T^4 = column 3 of T^(k+1):
column 0 of T^4 = column 3 of T = column 0 of T shift left;
column 1 of T^4 = column 3 of T^2;
column 2 of T^4 = column 3 of T^3.
PROG
(PARI) {T(n, k) = if(k>n||k<0, 0, if(k==n, 1, if(k==0, T(n+2, 3), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ); )}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) /* Column k of T = column 0 of T^(k+1): */
{T(n, k) = local(M=if(n==0, Mat(1), matrix(n, n, r, c, if(r>=c, T(r-1, c-1))))); if(k==n, 1, if(k==0, (M^4)[n, 1], (M^(k+1))[n-k+1, 1]))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 11 2008
STATUS
approved