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.

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.

Links

Table of n, a(n) for n=0..59.

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

Adjacent sequences: A109313 A109314 A109315 * A109317 A109318 A109319

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 07 2005

Status

approved