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Search: id:A132623
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| A132623 |
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Triangle T, read by rows, where T(n,k) = Sum_{j=1..n-k-1} [T^j](n-1,k) with T(n+1,n) = n+1 and T(n,n)=0 for n>=0, where T^n denotes the n-th matrix power of T. |
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+0
2
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| 0, 1, 0, 1, 2, 0, 3, 2, 3, 0, 14, 8, 3, 4, 0, 87, 46, 15, 4, 5, 0, 669, 338, 102, 24, 5, 6, 0, 6098, 2992, 861, 188, 35, 6, 7, 0, 64050, 30800, 8589, 1788, 310, 48, 7, 8, 0, 759817, 360110, 98238, 19800, 3275, 474, 63, 8, 9, 0, 10028799, 4701734, 1262208, 248624
(list; table; graph; listen)
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OFFSET
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0,5
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EXAMPLE
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Triangle begins:
0;
1, 0;
1, 2, 0;
3, 2, 3, 0;
14, 8, 3, 4, 0;
87, 46, 15, 4, 5, 0;
669, 338, 102, 24, 5, 6, 0;
6098, 2992, 861, 188, 35, 6, 7, 0;
64050, 30800, 8589, 1788, 310, 48, 7, 8, 0;
759817, 360110, 98238, 19800, 3275, 474, 63, 8, 9, 0; ...
MATRIX POWER SERIES PROPERTY.
[I - T]^-1 = Sum_{n>=0} T^n and equals T shifted up 1 row
(with '1's in the main diagonal):
1;
1, 1;
3, 2, 1;
14, 8, 3, 1;
87, 46, 15, 4, 1;
669, 338, 102, 24, 5, 1; ...
GENERATE T FROM MATRIX POWERS OF T.
Matrix square T^2 begins:
0;
0, 0;
2, 0, 0;
5, 6, 0, 0;
23, 14, 12, 0, 0;
143, 78, 27, 20, 0, 0; ...
so that T(4,1) = T(3,1) + [T^2](3,1) = 2 + 6 = 8;
and T(3,0) = T(2,0) + [T^2](2,0) = 1 + 2 = 3.
Matrix cube T^3 begins:
0;
0, 0;
0, 0, 0;
6, 0, 0, 0;
26, 24, 0, 0, 0;
165, 94, 60, 0, 0, 0; ...
so that T(5,1) = T(4,1) + [T^2](4,1) + [T^3](4,1) = 8 + 14 + 24 = 46;
and T(4,0) = T(3,0) + [T^2](3,0) + [T^3](3,0) = 3 + 5 + 6 = 14.
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PROGRAM
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(PARI) {T(n, k)=local(M=if(n<=0, Mat(1), matrix(n, n, r, c, if(r>=c, T(r-1, c-1))))); if(n<k|k<0, 0, if(n==k, 0, if(n==k+1, n, sum(j=1, n-k-1, (M^j)[n, k+1]) )))}
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CROSSREFS
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Cf. A132624 (column 0).
Sequence in context: A100949 A110493 A118234 this_sequence A051613 A077961 A077962
Adjacent sequences: A132620 A132621 A132622 this_sequence A132624 A132625 A132626
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Aug 25 2007
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