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 A059365 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. 29

%I

%S 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,

%T 132,90,48,20,6,1,0,429,429,297,165,75,27,7,1,0,1430,1430,1001,572,

%U 275,110,35,8,1,0,4862,4862,3432,2002,1001,429,154,44

%N 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.

%H G. C. Greubel, <a href="/A059365/b059365.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H F. R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., 204 (1999), 73-112.

%H D. Callan, <a href="http://arXiv.org/abs/math.CO/0211380">A recursive bijective approach to counting permutations containing 3-letter patterns</a>, arXiv:math/0211380 [math.CO], 2002.

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000011">The number of touch points of a Dyck path.</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000025">The number of initial rises of a Dyck paths.</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000061">The number of nodes on the left branch of the tree.</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000084">The number of subtrees.</a>

%H A. Robertson, D. Saracino and D. Zeilberger, <a href="http://arXiv.org/abs/math.CO/0203033">Refined restricted permutations</a>, arXiv:math/0203033 [math.CO], 2002.

%F Essentially the same triangle as [0, 1, 1, 1, 1, 1, 1, ...] DELTA A000007, where DELTA is DelĂ©ham's operator defined in A084938, but the first term is T(0,0) = 0.

%e Triangle starts

%e 0;

%e 0, 1;

%e 0, 1, 1;

%e 0, 2, 2, 1;

%e 0, 5, 5, 3, 1;

%e 0, 14, 14, 9, 4, 1;

%e 0, 42, 42, 28, 14, 5, 1;

%e 0, 132, 132, 90, 48, 20, 6, 1;

%e 0, 429, 429, 297, 165, 75, 27, 7, 1;

%e 0, 1430, 1430, 1001, 572, 275, 110, 35, 8, 1;

%e 0, 4862, 4862, 3432, 2002, 1001, 429, 154, 44, 9, 1;

%e ...

%t 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 *)

%o (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

%o (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

%Y See also the triangle in A009766. First 2 diagonals both give A000108, next give A000245, A002057.

%Y Cf. A009766 A000007 A084938 A000108.

%Y The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.

%Y Essentially the same as A033184.

%Y The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

%Y Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...

%K nonn,tabl

%O 0,8

%A _N. J. A. Sloane_, Jan 28 2001

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Last modified January 17 14:12 EST 2019. Contains 319225 sequences. (Running on oeis4.)