0,5

Here SHIFT_RIGHT(T) shifts the columns of T one place to the right and fills column 0 with [1,0,0,0,...].

T(n,2k+1) = Sum_{j=0..n-2k-1} T(n-k-1,k+j)*T(k+j,k) for n>2k and T(n,2k) = Sum_{j=0..n-2k} T(n-k,k+j)*T(k-1+j,k-1) for n>=2k, with T(n,n)=T(n,0)=1.

Triangle T begins:

1;

1, 1;

1, 2, 1;

1, 4, 3, 1;

1, 9, 8, 4, 1;

1, 23, 22, 14, 5, 1;

1, 66, 65, 50, 20, 6, 1;

1, 209, 208, 191, 79, 28, 7, 1;

1, 724, 723, 780, 322, 126, 37, 8, 1;

1, 2722, 2721, 3415, 1385, 572, 180, 48, 9, 1;

1, 11054, 11053, 15924, 6293, 2692, 871, 264, 58, 10, 1; ...

The matrix square T^2 = A117427:

1;

2, 1;

4, 4, 1;

9, 14, 6, 1;

23, 50, 28, 8, 1;

66, 191, 126, 48, 10, 1;

209, 780, 572, 264, 70, 12, 1; ...

where column k of T^2 equals column 2k+1 of T.

Let matrix R = SHIFT_RIGHT(T):

1;

0, 1;

0, 1, 1;

0, 1, 2, 1;

0, 1, 4, 3, 1;

0, 1, 9, 8, 4, 1;

0, 1, 23, 22, 14, 5, 1; ...

then matrix product T*R = A117425:

1;

1, 1;

1, 3, 1;

1, 8, 5, 1;

1, 22, 20, 7, 1;

1, 65, 79, 37, 9, 1;

1, 208, 322, 180, 58, 11, 1; ...

where column k of T*R equals column 2k of T.

(PARI) {T(n, k)=if(n<k|k<0, 0, if(n==k|k==0, 1, if(n==k+1, n, sum(j=0, n-k, T(n-((k+1)\2), k\2+j)*T((k-1)\2+j, (k-1)\2)))))}

Cf. A117425 (T*SHIFT_RIGHT(T)), A117427 (T^2); columns: A117419, A117420, A117421, A117422, A117423, A117424.

Sequence in context: A033185 A105632 A091491 this_sequence A101494 A125781 A091150

Adjacent sequences: A117415 A117416 A117417 this_sequence A117419 A117420 A117421

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Mar 14 2006