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A132844
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Triangle, read by rows, where T(n,k) = {T^[(n+k)/2]}( [(n+k)/2], k) for n>=k>=0, so that antidiagonal {2n} equals row n of T^n for n>=0 and odd antidiagonals equal even antidiagonals.
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5
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 3, 1, 1, 3, 9, 3, 4, 1, 1, 13, 9, 18, 4, 5, 1, 1, 13, 42, 18, 30, 5, 6, 1, 1, 73, 42, 95, 30, 45, 6, 7, 1, 1, 73, 270, 95, 179, 45, 63, 7, 8, 1, 1, 466, 270, 693, 179, 301, 63, 84, 8, 9, 1, 1, 466, 1785, 693, 1463, 301, 468, 84, 108, 9, 10, 1, 1
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OFFSET
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0,8
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COMMENTS
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Column k of triangle A132845 is equal to column k of this triangle but without repetition of terms.
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LINKS
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Table of n, a(n) for n=0..77.
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FORMULA
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T(n,k) = A132845( [(n+k)/2], k) where A132845(n,k) = [T^n](n,k) for n>=k>=0.
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EXAMPLE
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Triangle T begins:
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
3, 2, 3, 1, 1;
3, 9, 3, 4, 1, 1;
13, 9, 18, 4, 5, 1, 1;
13, 42, 18, 30, 5, 6, 1, 1;
73, 42, 95, 30, 45, 6, 7, 1, 1;
73, 270, 95, 179, 45, 63, 7, 8, 1, 1;
466, 270, 693, 179, 301, 63, 84, 8, 9, 1, 1;
466, 1785, 693, 1463, 301, 468, 84, 108, 9, 10, 1, 1;
3309, 1785, 4893, 1463, 2726, 468, 687, 108, 135, 10, 11, 1, 1; ...
Matrix square T^2 begins:
1;
2, 1;
3, 2, 1; <-- antidiagonals 4, 5, of T
5, 5, 2, 1;
12, 9, 7, 2, 1;
25, 31, 13, 9, 2, 1;
75, 63, 58, 17, 11, 2, 1; ...
Matrix cube T^3 begins:
1;
3, 1;
6, 3, 1;
13, 9, 3, 1; <-- antidiagonals 6, 7, of T
33, 22, 12, 3, 1;
87, 75, 31, 15, 3, 1;
265, 204, 132, 40, 18, 3, 1; ...
Matrix 4-th power T^4 begins:
1;
4, 1;
10, 4, 1;
26, 14, 4, 1;
73, 42, 18, 4, 1; <-- antidiagonals 8, 9, of T
220, 151, 58, 22, 4, 1;
717, 488, 253, 74, 26, 4, 1; ...
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PROG
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(PARI) {T(n, k)=local(M=matrix((n+k)\2+1, (n+k)\2+1, r, c, if(r>=c, if(r<=c+1, 1, T(r-1, c-1))))); (M^((n+k)\2))[(n+k)\2+1, k+1]}
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CROSSREFS
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Cf. A132845 (triangle); columns: A132846, A132847, A132848, A132849.
Sequence in context: A104467 A132463 A153901 * A006843 A049456 A117506
Adjacent sequences: A132841 A132842 A132843 * A132845 A132846 A132847
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna, Sep 17 2007
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STATUS
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approved
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