A128320 Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n that are to the right of T(n,k) with the vector of terms in column k that are above T(n,k) for n>k+1>0, with the odd numbers in the secondary diagonal and all 1's in the main diagonal.

1, 1, 1, 4, 3, 1, 17, 8, 5, 1, 98, 41, 12, 7, 1, 622, 234, 73, 16, 9, 1, 4512, 1602, 418, 113, 20, 11, 1, 35373, 11976, 3110, 650, 161, 24, 13, 1, 300974, 98541, 23920, 5242, 930, 217, 28, 15, 1, 2722070, 866942, 207549, 41304, 8094, 1258, 281, 32, 17, 1

Offset

0,4

Links

Table of n, a(n) for n=0..54.

Formula

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

Example

Illustrate the recurrence by:
T(n,k)=[T(n,k+1),T(n,k+2),..,T(n,n)]*[T(k,k),T(k+1,k),..,T(n-1,k)]:
T(3,0) = [8,5,1]*[1,1,4]~ = 8*1 + 5*1 + 1*4 = 17;
T(4,1) = [12,7,1]*[1,3,8]~ = 12*1 + 7*3 + 1*8 = 41;
T(5,1) = [73,16,9,1]*[1,3,8,41]~ = 73*1 + 16*3 + 9*8 + 1*41 = 234;
T(6,3) = [113,20,11,1]*[1,5,12,73]~ = 113*1 + 20*5 + 11*12 + 1*73 = 418.
Triangle begins:
1;
1, 1;
4, 3, 1;
17, 8, 5, 1;
98, 41, 12, 7, 1;
622, 234, 73, 16, 9, 1;
4512, 1602, 418, 113, 20, 11, 1;
35373, 11976, 3110, 650, 161, 24, 13, 1;
300974, 98541, 23920, 5242, 930, 217, 28, 15, 1;
2722070, 866942, 207549, 41304, 8094, 1258, 281, 32, 17, 1;
26118056, 8139602, 1885166, 377757, 65088, 11762, 1634, 353, 36, 19, 1;

Prog

(PARI) {T(n, k)=if(n==k, 1, if(n==k+1, 2*n-1, sum(i=0, n-k-1, T(n, k+i+1)*T(k+i, k))))}

Crossrefs

Cf. A128321 (column 0), A128322 (column 1), A128323 (column 2), A128324 (row sums); variant: A115080.

Sequence in context: A098234 A193795 A181355 * A189507 A348436 A350528

Adjacent sequences: A128317 A128318 A128319 * A128321 A128322 A128323

Keyword

nonn,tabl

Author

Paul D. Hanna, Feb 25 2007

Status

approved