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A050157
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T(n, k) = S(2n, n, k) for 0<=k<=n and n>=0, where S(p, q, r) is the number of upright paths from (0, 0) to (p, p-q) that do not rise above the line y = x-r.
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13
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1, 1, 2, 2, 5, 6, 5, 14, 19, 20, 14, 42, 62, 69, 70, 42, 132, 207, 242, 251, 252, 132, 429, 704, 858, 912, 923, 924, 429, 1430, 2431, 3068, 3341, 3418, 3431, 3432, 1430, 4862, 8502, 11050, 12310, 12750, 12854, 12869, 12870
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OFFSET
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0,3
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COMMENTS
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Let V = (e(1),...,e(n)) consist of q 1's and p-q 0's; let V(h) = (e(1),...,e(h)) and m(h) = (#1's in V(h)) - (#0's in V(h)) for h=1,...,n. Then S(p,q,r) is the number of V having r >= max{m(h)}.
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LINKS
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FORMULA
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T(n, k) = Sum_{0<=j<=k} t(n, j), array t as in A039599.
T(n, k) = binomial(2*n, n) - binomial(2*n, n+k+1). - Peter Luschny, Dec 21 2017
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EXAMPLE
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The triangle starts:
1
1, 2
2, 5, 6
5, 14, 19, 20
14, 42, 62, 69, 70
42, 132, 207, 242, 251, 252
132, 429, 704, 858, 912, 923, 924
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MAPLE
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A050157 := (n, k) -> binomial(2*n, n) - binomial(2*n, n+k+1):
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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