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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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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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Array starts:
1 1 2 5 14 42 132 429
0 1 3 9 28 90 297 1001
1 2 6 19 62 207 704 2431
1 4 13 43 145 497 1727 6071
3 9 30 102 352 1230 4344 15483
6 21 72 250 878 3114 11139 40143
15 51 178 628 2236 8025 29004 105477
36 127 450 1608 5789 20979 76473 280221
91 323 1158 4181 15190 55494 203748 751422
232 835 3023 11009 40304 148254 547674 2031054
603 2188 7986 29295 107950 399420 1483380 5527750
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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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);
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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)
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CROSSREFS
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KEYWORD
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AUTHOR
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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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