A331322 a(n) = (3*n + 1)!/(n!)^3.

1, 24, 630, 16800, 450450, 12108096, 325909584, 8779605120, 236637794250, 6380456082000, 172080900531540, 4641917845743360, 125235075213284400, 3379123922914656000, 91184624634161304000, 2460769070127233057280, 66411927755894739034170, 1792432652235221330334000

Offset: 0

Author

Peter Luschny, Jan 18 2020

Keywords

• nonn

Comments

Diagonal of the rational function 1 / (1 - x - y - z)^2. - Ilya Gutkovskiy, Apr 23 2025

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A000108, A006480, A117671, A331431.

Programs

Magma

[(n+1)^2*Binomial(3*n+1,n+1)*Catalan(n): n in [0..25]]; // G. C. Greubel, Mar 22 2022

Maple

a := n -> (3*n+1)!/(n!)^3: seq(a(n), n=0..17); # Or:
hypergeom([2/3, 4/3], [1], 27*x): ser := series(%, x, 20):
seq(coeff(%, x, n), n=0..17); # Or:
a := proc(n) option remember; if n=0 then 1 else 3*(9 - n^(-2))*a(n-1) fi end:
# 4th Maple program:
W:=proc(x)sqrt(3)*MeijerG([[], [0, 0]], [[1/3, -1/3], []], x/27)/(18*Pi);end;
a:=proc(n) round(evalf[32](int(x^n*W(x),x=0..27)));end;
seq(a(n),n=0..17);
# Karol A. Penson, Jul 28 2023

Mathematica

Table[(3*n+1)*Binomial[3*n,n]*Binomial[2*n,n], {n,0,25}] (* G. C. Greubel, Mar 22 2022 *)

Sage

[(3*n+1)*binomial(2*n,n)*binomial(3*n,n) for n in (0..25)] # G. C. Greubel, Mar 22 2022

Formula

a(n) = [x^n] hypergeom([2/3, 4/3], [1], 27*x).

a(n) = 3*(9 - n^(-2))*a(n-1) for n > 0.

a(n) = (-1)^n*A331431(2*n, n).

a(n) = (n+1)^2*A117671(n)*A000108(n). - G. C. Greubel, Mar 22 2022

From Karol A. Penson, Jul 28 2023: (Start)

a(n) = Integral_{x=0..27} x^n*W(x) dx, where the weight function W(x) is defined on (0, 27) and it can be expressed with the Meijer G-function MeijerG as: W(x) = (sqrt(3)/(18*Pi))*MeijerG([[],[0,0]],[[-1/3,1/3],[]],x/27). The function W(x) is positive on its support (0, 27), is singular at x=0, and decreases monotonically to zero at x = 27.

The function W(x) is unique as it is the solution of the Hausdorff moment problem with the moments a(n). Due to the presence of two equal parameters (0,0) in MeijerG, it is not certain if W(x) can be represented by other known special functions. (End)

From Peter Bala, Oct 10 2024: (Start)

a(n) = (3*n + 1)*A006480(n).

a(n-1) = 1/(8*n^3) * Sum_{k = 0..2*n} (-1)^(n+k) * k*(2*n-k)^3 * binomial(2*n, k)^3 for n >= 1.

a(n-1) = 1/(4*n^2) * Sum_{k = 0..2*n-1} (-1)^(n+k) * k^3 * binomial(2*n, k)^2 * binomial(2*n-1, k) for n >= 1. (End)