A098446 Triangle, read by rows, such that T(n,k) equals the k-th term of the convolution of the (n-1)-th diagonal with the k-th row of this triangle.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 9, 1, 1, 5, 16, 30, 24, 1, 1, 6, 25, 70, 115, 77, 1, 1, 7, 36, 135, 344, 510, 295, 1, 1, 8, 49, 231, 805, 1908, 2602, 1329, 1, 1, 9, 64, 364, 1616, 5325, 11904, 15133, 6934, 1, 1, 10, 81, 540, 2919, 12381, 39001, 83028, 99367, 41351, 1

Offset

0,5

Comments

The rows of this triangle are the reverse of the rows of triangle A091351, in which the k-th column lists the row sums of the k-th matrix power of A091351. Row sums form A091352 and equal the secondary diagonal.

Links

Table of n, a(n) for n=0..65.

Formula

T(n, k) = Sum_{i=0..k} T(k, i)*T(n-i-1, k-i) for 0<k<n, else T(0, n)=T(n, n)=1.

Example

T(7,3) = T(3,0)*T(6,3) + T(3,1)*T(5,2) + T(3,2)*T(4,1) + T(3,3)*T(3,0)
= 1*70 + 3*16 + 4*4 + 1*1 = 135.
Rows begin:
[1],
[1,1],
[1,2,1],
[1,3,4,1],
[1,4,9,9,1],
[1,5,16,30,24,1],
[1,6,25,70,115,77,1],
[1,7,36,135,344,510,295,1],
[1,8,49,231,805,1908,2602,1329,1],
[1,9,64,364,1616,5325,11904,15133,6934,1],...

Prog

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

Crossrefs

Cf. A091351, A091352.

Sequence in context: A009999 A322268 A144823 * A098447 A202784 A175105

Adjacent sequences: A098443 A098444 A098445 * A098447 A098448 A098449

Keyword

nonn,tabl

Author

Paul D. Hanna, Sep 07 2004

Status

approved