login
A109316
Triangle T, read by rows, where T(n,k) = [T^2](n-1,k) + [T^2](n-2,k-1) (n>k>0), with T(n,0) = [T^2](n-1,0) (n>0) and T(n,n) = 1 (n>=0), where T^2 is the matrix square of T.
3
1, 1, 1, 2, 2, 1, 6, 6, 2, 1, 22, 22, 8, 2, 1, 94, 94, 36, 8, 2, 1, 446, 446, 176, 40, 8, 2, 1, 2294, 2294, 920, 216, 40, 8, 2, 1, 12542, 12542, 5080, 1224, 224, 40, 8, 2, 1, 71974, 71974, 29336, 7200, 1328, 224, 40, 8, 2, 1, 429342, 429342, 175752, 43712, 8160
OFFSET
0,4
COMMENTS
Limit of rows read backwards = {1,2,8,40,224,...,2^n*A000108(n),...} where A000108(n) = C(2n,n)/(n+1) is the n-th Catalan number.
EXAMPLE
Triangle T begins:
1;
1,1;
2,2,1;
6,6,2,1;
22,22,8,2,1;
94,94,36,8,2,1;
446,446,176,40,8,2,1;
2294,2294,920,216,40,8,2,1;
12542,12542,5080,1224,224,40,8,2,1;
71974,71974,29336,7200,1328,224,40,8,2,1; ...
Matrix square T^2 starts:
1;
2,1;
6,4,1;
22,16,4,1;
94,72,20,4,1;
446,352,104,20,4,1;
2294,1848,568,112,20,4,1; ...
where T(n,k) = [T^2](n-1,k) + [T^2](n-2,k-1):
T(5,2) = [T^2](4,2) + [T^2](3,1) = 20 + 16 = 36;
T(7,3) = [T^2](6,3) + [T^2](5,2) = 112 + 104 = 216.
PROG
(PARI) T(n, k)=local(M=matrix(n, n, r, c, if(r>=c, T(r-1, c-1)))); if(n<k || k<0, 0, if(n==k || n<=1, 1, (M^2)[n, k+1]+if(k>0, (M^2)[n-1, k])))
CROSSREFS
Cf. A109317 (column 0), A109318 (column 1), A109319 (row sums), A000108.
Sequence in context: A344639 A230420 A137381 * A162980 A162979 A094587
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jul 07 2005
STATUS
approved