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 A059346 Difference array of Catalan numbers A000108 read by antidiagonals. 9
 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, 36, 51, 72, 102, 145, 207, 297, 429, 91, 127, 178, 250, 352, 497, 704, 1001, 1430, 232, 323, 450, 628, 878, 1230, 1727, 2431, 3432, 4862, 603, 835, 1158, 1608, 2236, 3114 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112. Zhousheng Mei, Suijie Wang, Pattern Avoidance of Generalized Permutations, arXiv:1804.06265 [math.CO], 2018. Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29. FORMULA 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 EXAMPLE Triangle starts:   1;   0,  1;   1,  1,  2;   1,  2,  3,  5;   3,  4,  6,  9, 14; MAPLE # Uses floating point, precision might have to be adjusted. C := n -> binomial(2*n, n)/(n+1); H := (n, k) -> hypergeom([k-n, k+1/2], [k+2], 4); T := (n, k) -> (-1)^(n-k)*C(k)*H(n, k); seq(print(seq(round(evalf(T(n, k), 32)), k=0..n)), n=0..7); # Peter Luschny, Aug 16 2012 MATHEMATICA 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 *) PROG (Sage) def T(n, k) :     if k > n : return 0     if n == k : return binomial(2*n, n)/(n+1)     return T(n-1, k) - T(n, k+1) A059346 = lambda n, k: (-1)^(n-k)*T(n, k) for n in (0..5): [A059346(n, k) for k in (0..n)] # Peter Luschny, Aug 16 2012 CROSSREFS Top row is A000108, leading diagonals give A005043, A001006, A005554. Row sums are A106640. Cf. A000108, A000245, A026012, A033434, A106534. Sequence in context: A117673 A107946 A054502 * A259439 A274491 A076492 Adjacent sequences:  A059343 A059344 A059345 * A059347 A059348 A059349 KEYWORD nonn,easy,nice,tabl AUTHOR N. J. A. Sloane, Jan 27 2001 EXTENSIONS More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001 STATUS approved

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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)