A125278 Convolution triangle of A030266, which shifts left under self-COMPOSE.

1, 1, 1, 2, 2, 1, 6, 5, 3, 1, 23, 16, 9, 4, 1, 104, 62, 31, 14, 5, 1, 531, 278, 123, 52, 20, 6, 1, 2982, 1398, 552, 213, 80, 27, 7, 1, 18109, 7718, 2750, 964, 340, 116, 35, 8, 1, 117545, 46083, 14976, 4784, 1561, 513, 161, 44, 9, 1, 808764, 294392, 88083, 25792, 7755

Offset

0,4

Comments

Column 0 of matrix square T^2 equals column 0 of T shift left. Central terms are T(2*n,n) = (n+1)*A125279(n).

Links

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

Formula

T(0,0) = 1 ; for n>0: T(n,0) = Sum_{j=0..n-1} T(j,0)*T(n-1,j) = A030266(n) (self-COMPOSE); for k>0: T(n,k) = Sum_{j=0..n-k} T(j,0)*T(n-1-j,k-1) (self-convolutions of A030266).

Example

Triangle begins:
1;
1, 1;
2, 2, 1;
6, 5, 3, 1;
23, 16, 9, 4, 1;
104, 62, 31, 14, 5, 1;
531, 278, 123, 52, 20, 6, 1;
2982, 1398, 552, 213, 80, 27, 7, 1;
18109, 7718, 2750, 964, 340, 116, 35, 8, 1;
117545, 46083, 14976, 4784, 1561, 513, 161, 44, 9, 1;
808764, 294392, 88083, 25792, 7755, 2400, 742, 216, 54, 10, 1; ...
Matrix square T^2 begins:
1;
2, 1;
6, 4, 1;
23, 16, 6, 1;
104, 70, 30, 8, 1;
531, 336, 149, 48, 10, 1; ...
which is also a convolution triangle.

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k, 1, if(k>0, sum(j=0, n-k, T(j, 0)*T(n-1-j, k-1)), sum(j=0, n-1, T(j, 0)*T(n-1, j)))))

Crossrefs

Cf. A030266, A125279, A125280 (row sums).

Sequence in context: A271776 A171486 A127743 * A134558 A344639 A230420

Adjacent sequences: A125275 A125276 A125277 * A125279 A125280 A125281

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 26 2006

Status

approved