A096591 Triangle T, read by rows, such that the matrix square shifts T one place diagonally left and upward, with T(n,0)=T(n,n)=1 for n>=0.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 5, 5, 2, 1, 1, 1, 8, 10, 5, 2, 1, 1, 1, 15, 22, 14, 5, 2, 1, 1, 1, 28, 47, 34, 14, 5, 2, 1, 1, 1, 61, 113, 88, 42, 14, 5, 2, 1, 1, 1, 133, 269, 223, 116, 42, 14, 5, 2, 1, 1, 1, 328, 705, 609, 333, 132, 42, 14, 5, 2, 1, 1, 1, 807, 1843, 1660

Offset

0,8

Comments

Column with index 1 forms the row sums shift right. The convergent of the rows in reverse order is the Catalan sequence (A000108).

Links

Table of n, a(n) for n=0..81.

Formula

T(n, k) = Sum_{i=0..n-1} T(n-2, i)*T(i, k-1) for n>1 and k>0; T(n, 0)=T(n, n)=1.

Example

Rows begin:
[1],
[1,1],
[1,1,1],
[1,2,1,1],
[1,3,2,1,1],
[1,5,5,2,1,1],
[1,8,10,5,2,1,1],
[1,15,22,14,5,2,1,1],
[1,28,47,34,14,5,2,1,1],
[1,61,113,88,42,14,5,2,1,1],
[1,133,269,223,116,42,14,5,2,1,1],...
The matrix square of T is given by:
[1],
[2,1],
[3,2,1],
[5,5,2,1],
[8,10,5,2,1],
[15,22,14,5,2,1],
[28,47,34,14,5,2,1],
[61,113,88,42,14,5,2,1],
[133,269,223,116,42,14,5,2,1],...
which equals T shift one place diagonally left and upward.

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k || k==0, 1, sum(i=0, n-1, T(n-2, i)*T(i, k-1)); ))

Crossrefs

Cf. A096592, A000108.

Sequence in context: A225630 A129713 A096669 * A316074 A332954 A115568

Adjacent sequences: A096588 A096589 A096590 * A096592 A096593 A096594

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 28 2004

Status

approved