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A059365
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Another version of the Catalan triangle: T(r,s) = binomial(2*r-s-1,r-1) - binomial(2*r-s-1,r), r>=0, 0 <= s <= r.
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30
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0, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 3, 1, 0, 14, 14, 9, 4, 1, 0, 42, 42, 28, 14, 5, 1, 0, 132, 132, 90, 48, 20, 6, 1, 0, 429, 429, 297, 165, 75, 27, 7, 1, 0, 1430, 1430, 1001, 572, 275, 110, 35, 8, 1, 0, 4862, 4862, 3432, 2002, 1001, 429, 154, 44
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OFFSET
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0,8
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LINKS
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FORMULA
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Essentially the same triangle as [0, 1, 1, 1, 1, 1, 1, ...] DELTA A000007, where DELTA is Deléham's operator defined in A084938, but the first term is T(0,0) = 0.
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EXAMPLE
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Triangle starts
0;
0, 1;
0, 1, 1;
0, 2, 2, 1;
0, 5, 5, 3, 1;
0, 14, 14, 9, 4, 1;
0, 42, 42, 28, 14, 5, 1;
0, 132, 132, 90, 48, 20, 6, 1;
0, 429, 429, 297, 165, 75, 27, 7, 1;
0, 1430, 1430, 1001, 572, 275, 110, 35, 8, 1;
0, 4862, 4862, 3432, 2002, 1001, 429, 154, 44, 9, 1;
...
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MATHEMATICA
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Table[Binomial[2*r - s - 1, r - 1] - Binomial[2*r - s - 1, r], {r, 0, 10}, {s, 0, r}] // Flatten (* G. C. Greubel, Jan 08 2017 *)
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PROG
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(PARI) tabl(nn) = { print(0); for (r=1, nn, for (s=0, r, print1(binomial(2*r-s-1, r-1)-binomial(2*r-s-1, r), ", "); ); print(); ); } \\ Michel Marcus, Nov 01 2013
(Magma) /* as triangle */ [[[0] cat [Binomial(2*r-s-1, r-1)- Binomial(2*r-s-1, r): s in [1..r]]: r in [0..10]]]; // Vincenzo Librandi, Jan 09 2017
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CROSSREFS
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The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A053121, A059365, A099039, A106566, A130020, A047072, A171567, A181645.
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KEYWORD
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AUTHOR
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STATUS
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approved
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