A356546 Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
1, 2, 2, 6, 12, 6, 20, 60, 60, 20, 70, 280, 420, 280, 70, 252, 1260, 2520, 2520, 1260, 252, 924, 5544, 13860, 18480, 13860, 5544, 924, 3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432, 12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870
Offset: 0
Examples
Triangle T(n, k) begins: [0] 1; [1] 2, 2; [2] 6, 12, 6; [3] 20, 60, 60, 20; [4] 70, 280, 420, 280, 70; [5] 252, 1260, 2520, 2520, 1260, 252; [6] 924, 5544, 13860, 18480, 13860, 5544, 924; [7] 3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432; [8] 12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870;
Crossrefs
Programs
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Maple
A356546 := (n, k) -> pochhammer(n+1, n)/(k!*(n-k)!): for n from 0 to 8 do seq(A356546(n, k), k=0..n) od;
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Mathematica
T[ n_, k_] := Binomial[2*n, n] * Binomial[n, k]; (* Michael Somos, Aug 18 2022 *)
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PARI
{T(n, k) = binomial(2*n, n) * binomial(n, k)}; /* Michael Somos, Aug 18 2022 */ -
SageMath
def A356546(n, k): return rising_factorial(n+1,n) // (factorial(k) * factorial(n-k)) for n in range(9): print([A356546(n, k) for k in range(n+1)])
Formula
Bernoulli(n) / Catalan(n) = Sum_{k=0..n} (-1)^k*A173018(n, k) / T(n, k), (with Bernoulli(1) = 1/2).
G.f.: 1/sqrt(1 - 4*x*(y + 1)). - Vladimir Kruchinin, Feb 15 2023
Comments