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A356546 Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).

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