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Search: id:A109282
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| A109282 |
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Triangle T, read by rows, that satisfies: T(n,k) = [T^3](n-1,k) for n>k+1>=1, with T(n,n) = 1 and T(n+1,n) = n+1 for n>=0, where T^3 is the matrix cube of T. |
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+0
5
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| 1, 1, 1, 3, 2, 1, 15, 6, 3, 1, 96, 36, 9, 4, 1, 735, 258, 63, 12, 5, 1, 6447, 2190, 492, 96, 15, 6, 1, 63120, 20988, 4545, 804, 135, 18, 7, 1, 677739, 222042, 46935, 7980, 1200, 180, 21, 8, 1, 7878921, 2554890, 530562, 87960, 12675, 1686, 231, 24, 9, 1
(list; table; graph; listen)
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OFFSET
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0,4
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FORMULA
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T^(m+3) = SHIFT_UP(T^(m+1) - T^m) - D*T^m for all m where diagonal matrix D = [0, 1, 2, 3, ...] and SHIFT_UP shifts each column up 1 row.
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EXAMPLE
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Triangle T begins:
1;
1,1;
3,2,1;
15,6,3,1;
96,36,9,4,1;
735,258,63,12,5,1;
6447,2190,492,96,15,6,1;
63120,20988,4545,804,135,18,7,1;
677739,222042,46935,7980,1200,180,21,8,1; ...
Matrix cube T^3 starts:
1;
3,1;
15,6,1;
96,36,9,1;
735,258,63,12,1;
6447,2190,492,96,15,1; ...
which equals SHIFT_UP(T) - D where
D is the diagonal matrix [0,1,2,3,...].
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PROGRAM
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(PARI) {T(n, k)=local(M=matrix(n+3, n+3)); M=M^0; for(i=1, n, M=matrix(n+3, n+3, r, c, if(r>=c, if(r==c, 1, if(r==c+1, c, (M^3)[r-1, c]))))); return((M^3)[n+1, k+1])}
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CROSSREFS
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Cf. A109152, A109283 (column 0), A109284 (column 1), A109285 (column 2), A109286 (row sums).
Sequence in context: A112911 A111548 A140709 this_sequence A135902 A135876 A136217
Adjacent sequences: A109279 A109280 A109281 this_sequence A109283 A109284 A109285
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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), Jun 24 2005
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