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A234950
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Borel's triangle read by rows: T(n,k) = Sum_{s=k..n} binomial(s,k)*C(n,s), where C(n,s) is an entry in Catalan's triangle A009766.
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6
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1, 2, 1, 5, 6, 2, 14, 28, 20, 5, 42, 120, 135, 70, 14, 132, 495, 770, 616, 252, 42, 429, 2002, 4004, 4368, 2730, 924, 132, 1430, 8008, 19656, 27300, 23100, 11880, 3432, 429, 4862, 31824, 92820, 157080, 168300, 116688, 51051, 12870, 1430
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: 1/x*(1-sqrt(1-4*x-4*x*y))/(1+2*y+sqrt(1-4*x-4*x*y)). - Vladimir Kruchinin, Sep 04 2018
T(n,k) = 2*binomial(2*n+1,n)*(n-k+1)*binomial(n+1,k)/((k+n+1)*(k+n+2)). - Peter Luschny, Sep 04 2018
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EXAMPLE
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Triangle begins:
1,
2, 1,
5, 6, 2,
14, 28, 20, 5,
42, 120, 135, 70, 14,
132, 495, 770, 616, 252, 42,
429, 2002, 4004, 4368, 2730, 924, 132,
1430, 8008, 19656, 27300, 23100, 11880, 3432, 429,
...
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MAPLE
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T := (n, k) -> 2*binomial(2*n+1, n)*(n-k+1)*binomial(n+1, k)/((k+n+1)*(k+n+2)):
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MATHEMATICA
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T[n_, k_] := 2 Binomial[2n+1, n] (n-k+1) Binomial[n+1, k]/((k+n+1)(k+n+2));
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PROG
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(Haskell)
a234950 n k = sum [a007318 s k * a009766 n s | s <- [k..n]]
a234950_row n = map (a234950 n) [0..n]
a234950_tabl = map a234950_row [0..]
(PARI) T(n, k) = sum(s=k, n, binomial(s, k)*binomial(n+s, n)*(n-s+1)/(n+1));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print(); ); \\ Michel Marcus, Sep 06 2015
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CROSSREFS
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The two borders give the Catalan numbers A000108.
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KEYWORD
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AUTHOR
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STATUS
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approved
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