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A059346
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Difference array of Catalan numbers A000108 read by antidiagonals.
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9
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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)
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OFFSET
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0,6
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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Triangle starts:
1;
0, 1;
1, 1, 2;
1, 2, 3, 5;
3, 4, 6, 9, 14;
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MAPLE
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# 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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy,nice,tabl
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AUTHOR
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N. J. A. Sloane, Jan 27 2001
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001
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STATUS
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approved
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