%I #24 Feb 15 2023 04:41:01
%S 1,2,2,6,12,6,20,60,60,20,70,280,420,280,70,252,1260,2520,2520,1260,
%T 252,924,5544,13860,18480,13860,5544,924,3432,24024,72072,120120,
%U 120120,72072,24024,3432,12870,102960,360360,720720,900900,720720,360360,102960,12870
%N Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
%C The counterpart using the falling factorial is Leibniz's Harmonic Triangle A003506.
%F Bernoulli(n) / Catalan(n) = Sum_{k=0..n} (-1)^k*A173018(n, k) / T(n, k), (with Bernoulli(1) = 1/2).
%F G.f.: 1/sqrt(1 - 4*x*(y + 1)). - _Vladimir Kruchinin_, Feb 15 2023
%e Triangle T(n, k) begins:
%e [0] 1;
%e [1] 2, 2;
%e [2] 6, 12, 6;
%e [3] 20, 60, 60, 20;
%e [4] 70, 280, 420, 280, 70;
%e [5] 252, 1260, 2520, 2520, 1260, 252;
%e [6] 924, 5544, 13860, 18480, 13860, 5544, 924;
%e [7] 3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432;
%e [8] 12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870;
%p A356546 := (n, k) -> pochhammer(n+1, n)/(k!*(n-k)!):
%p for n from 0 to 8 do seq(A356546(n, k), k=0..n) od;
%t T[ n_, k_] := Binomial[2*n, n] * Binomial[n, k]; (* _Michael Somos_, Aug 18 2022 *)
%o (SageMath)
%o def A356546(n, k):
%o return rising_factorial(n+1,n) // (factorial(k) * factorial(n-k))
%o for n in range(9): print([A356546(n, k) for k in range(n+1)])
%o (PARI) {T(n, k) = binomial(2*n, n) * binomial(n, k)}; /* _Michael Somos_, Aug 18 2022 */
%Y cf. A000984, A059304 (row sums, see also A343842), A265609 (rising factorial).
%Y Cf. A003506, A173018 (Eulerian numbers), A000108, A000897 (central terms).
%K sign,tabl
%O 0,2
%A _Peter Luschny_, Aug 12 2022