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A173424
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Triangle read by rows: T(n, k) = (2*n - 2*k)!*(2*k)!/(2^n*(n - k)!*k!).
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0
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1, 1, 1, 3, 1, 3, 15, 3, 3, 15, 105, 15, 9, 15, 105, 945, 105, 45, 45, 105, 945, 10395, 945, 315, 225, 315, 945, 10395, 135135, 10395, 2835, 1575, 1575, 2835, 10395, 135135, 2027025, 135135, 31185, 14175, 11025, 14175, 31185, 135135, 2027025, 34459425
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OFFSET
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0,4
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LINKS
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FORMULA
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T(n, k) = (1/Pi) * 2^n * Gamma(k + 1/2) * Gamma(n - k + 1/2).
T(n, k) = (2*n-1)!! * binomial(n, k) / binomial(2*n, 2*k). (End)
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EXAMPLE
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Triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 3, 1, 3;
[3] 15, 3, 3, 15;
[4] 105, 15, 9, 15, 105;
[5] 945, 105, 45, 45, 105, 945;
[6] 10395, 945, 315, 225, 315, 945, 10395;
[7] 135135, 10395, 2835, 1575, 1575, 2835, 10395, 135135;
[8] 2027025, 135135, 31185, 14175, 11025, 14175, 31185, 135135, 2027025;
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MAPLE
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T := (n, k) -> doublefactorial(2*n-1) * binomial(n, k) / binomial(2*n, 2*k):
for n from 0 to 8 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Apr 15 2023
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MATHEMATICA
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t[n_, k_] = (2*n - 2*k)!*(2*k)!/(2^n*(n - k)!*k!);
Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}]; Flatten[%]
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Formula added by the Assoc. Editors of the OEIS, Feb 24 2010
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STATUS
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approved
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