A356546 - OEIS

%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