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A059346 Difference array of Catalan numbers A000108 read by antidiagonals.

%I A059346 #40 Jul 15 2024 10:36:19
%S A059346 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 A059346 36,51,72,102,145,207,297,429,91,127,178,250,352,497,704,1001,1430,
%U A059346 232,323,450,628,878,1230,1727,2431,3432,4862,603,835,1158,1608,2236,3114
%N A059346 Difference array of Catalan numbers A000108 read by antidiagonals.
%H A059346 G. C. Greubel, <a href="/A059346/b059346.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A059346 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 A059346 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 A059346 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.
%H A059346 Benjamin Testart, <a href="https://arxiv.org/abs/2407.07701">Completing the enumeration of inversion sequences avoiding one or two patterns of length 3</a>, arXiv:2407.07701 [math.CO], 2024. See p. 36.
%F A059346 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 A059346 Array starts:
%e A059346       1       1       2       5      14      42     132     429
%e A059346       0       1       3       9      28      90     297    1001
%e A059346       1       2       6      19      62     207     704    2431
%e A059346       1       4      13      43     145     497    1727    6071
%e A059346       3       9      30     102     352    1230    4344   15483
%e A059346       6      21      72     250     878    3114   11139   40143
%e A059346      15      51     178     628    2236    8025   29004  105477
%e A059346      36     127     450    1608    5789   20979   76473  280221
%e A059346      91     323    1158    4181   15190   55494  203748  751422
%e A059346     232     835    3023   11009   40304  148254  547674 2031054
%e A059346     603    2188    7986   29295  107950  399420 1483380 5527750
%e A059346 Triangle starts:
%e A059346   1;
%e A059346   0,  1;
%e A059346   1,  1,  2;
%e A059346   1,  2,  3,  5;
%e A059346   3,  4,  6,  9, 14;
%p A059346 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 A059346 # _Peter Luschny_, Aug 16 2012, updated May 25 2021
%t A059346 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 A059346 (Sage)
%o A059346 def T(n, k) :
%o A059346     if k > n : return 0
%o A059346     if n == k : return binomial(2*n, n)/(n+1)
%o A059346     return T(n-1, k) - T(n, k+1)
%o A059346 A059346 = lambda n,k: (-1)^(n-k)*T(n, k)
%o A059346 for n in (0..5): [A059346(n,k) for k in (0..n)] # _Peter Luschny_, Aug 16 2012
%Y A059346 Top row is A000108, leading diagonals give A005043, A001006, A005554.
%Y A059346 Row sums are A106640.
%Y A059346 Cf. A000108, A000245, A026012, A033434, A106534.
%K A059346 nonn,easy,nice,tabl
%O A059346 0,6
%A A059346 _N. J. A. Sloane_, Jan 27 2001
%E A059346 More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001