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%I #37 May 17 2023 05:29:48
%S 1,0,1,1,1,2,1,2,3,5,3,4,6,9,14,6,9,13,19,28,42,15,21,30,43,62,90,132,
%T 36,51,72,102,145,207,297,429,91,127,178,250,352,497,704,1001,1430,
%U 232,323,450,628,878,1230,1727,2431,3432,4862,603,835,1158,1608,2236,3114
%N Difference array of Catalan numbers A000108 read by antidiagonals.
%H G. C. Greubel, <a href="/A059346/b059346.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 Zhousheng Mei, Suijie Wang, <a href="https://arxiv.org/abs/1804.06265">Pattern Avoidance of Generalized Permutations</a>, arXiv:1804.06265 [math.CO], 2018.
%H Jocelyn Quaintance and Harris Kwong, <a href="http://www.emis.de/journals/INTEGERS/papers/n29/n29.Abstract.html">A combinatorial interpretation of the Catalan and Bell number difference tables</a>, Integers, 13 (2013), #A29.
%F T(n, k) = (-1)^(n-k)*binomial(2*k,k)/(k+1)*hypergeometric([k-n, k+1/2],[k+2], 4). - _Peter Luschny_, Aug 16 2012
%e Array starts:
%e 1 1 2 5 14 42 132 429
%e 0 1 3 9 28 90 297 1001
%e 1 2 6 19 62 207 704 2431
%e 1 4 13 43 145 497 1727 6071
%e 3 9 30 102 352 1230 4344 15483
%e 6 21 72 250 878 3114 11139 40143
%e 15 51 178 628 2236 8025 29004 105477
%e 36 127 450 1608 5789 20979 76473 280221
%e 91 323 1158 4181 15190 55494 203748 751422
%e 232 835 3023 11009 40304 148254 547674 2031054
%e 603 2188 7986 29295 107950 399420 1483380 5527750
%e Triangle starts:
%e 1;
%e 0, 1;
%e 1, 1, 2;
%e 1, 2, 3, 5;
%e 3, 4, 6, 9, 14;
%p T := (n,k) -> (-1)^(n-k)*binomial(2*k,k)*hypergeom([k-n,k+1/2], [k+2], 4)/(k+1): seq(seq(simplify(T(n,k)), k=0..n), n=0..10);
%p # _Peter Luschny_, Aug 16 2012, updated May 25 2021
%t max = 11; t = Table[ Differences[ Table[ CatalanNumber[k], {k, 0, max}], n], {n, 0, max}]; Flatten[ Table[t[[n-k+1, k]], {n, 1, max}, {k, 1, n}]] (* _Jean-François Alcover_, Nov 15 2011 *)
%o (Sage)
%o def T(n, k) :
%o if k > n : return 0
%o if n == k : return binomial(2*n, n)/(n+1)
%o return T(n-1, k) - T(n, k+1)
%o A059346 = lambda n,k: (-1)^(n-k)*T(n, k)
%o for n in (0..5): [A059346(n,k) for k in (0..n)] # _Peter Luschny_, Aug 16 2012
%Y Top row is A000108, leading diagonals give A005043, A001006, A005554.
%Y Row sums are A106640.
%Y Cf. A000108, A000245, A026012, A033434, A106534.
%K nonn,easy,nice,tabl
%O 0,6
%A _N. J. A. Sloane_, Jan 27 2001
%E More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001
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