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

0,2

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

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)

a(n) ~ 3^(3*n+3/2) / (2*Pi). - Amiram Eldar, Oct 16 2025

Maple

a := n -> (3*n+1)!/(n!)^3: seq(a(n), n=0..17);
# Alternative:
hypergeom([2/3, 4/3], [1], 27*x): ser := series(%, x, 20):
seq(coeff(%, x, n), n=0..17);
# Alternative:
a := proc(n) option remember; if n=0 then 1 else 3*(9 - n^(-2))*a(n-1) fi end:
# Alternative:
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 *)

Prog

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

Crossrefs

Cf. A000108, A006480, A117671, A331431.

Sequence in context: A358114 A097192 A182611 * A126153 A002553 A006201

Adjacent sequences: A331319 A331320 A331321 * A331323 A331324 A331325

Keyword

nonn,easy

Author

Peter Luschny, Jan 18 2020

Status

approved