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

0,2

Comments

The counterpart using the falling factorial is Leibniz's Harmonic Triangle A003506.

Links

Table of n, a(n) for n=0..44.

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

Example

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;

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;

Mathematica

T[ n_, k_] := Binomial[2*n, n] * Binomial[n, k]; (* Michael Somos, Aug 18 2022 *)

Prog

(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)])
(PARI) {T(n, k) = binomial(2*n, n) * binomial(n, k)}; /* Michael Somos, Aug 18 2022 */

Crossrefs

cf. A000984, A059304 (row sums, see also A343842), A265609 (rising factorial).

Cf. A003506, A173018 (Eulerian numbers), A000108, A000897 (central terms).

Sequence in context: A281351 A351081 A241669 * A178802 A156992 A285529

Adjacent sequences: A356543 A356544 A356545 * A356547 A356548 A356549

Keyword

sign,tabl

Author

Peter Luschny, Aug 12 2022

Status

approved