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A128320
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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.
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4
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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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.
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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;
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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))))}
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CROSSREFS
| Cf. A128321 (column 0), A128322 (column 1), A128323 (column 2), A128324 (row sums); variant: A115080.
Sequence in context: A098234 A193795 A181355 * A189507 A039621 A142158
Adjacent sequences: A128317 A128318 A128319 * A128321 A128322 A128323
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Feb 25 2007
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