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a(n) = (3*n + 1)!/(n!)^3.
2

%I #40 Oct 13 2024 07:07:31

%S 1,24,630,16800,450450,12108096,325909584,8779605120,236637794250,

%T 6380456082000,172080900531540,4641917845743360,125235075213284400,

%U 3379123922914656000,91184624634161304000,2460769070127233057280,66411927755894739034170,1792432652235221330334000

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

%H G. C. Greubel, <a href="/A331322/b331322.txt">Table of n, a(n) for n = 0..500</a>

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

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

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

%F a(n) = (n+1)^2*A117671(n)*A000108(n). - _G. C. Greubel_, Mar 22 2022

%F From _Karol A. Penson_, Jul 28 2023: (Start)

%F 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.

%F 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)

%F From _Peter Bala_, Oct 10 2024: (Start)

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

%F 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.

%F 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)

%p a := n -> (3*n+1)!/(n!)^3: seq(a(n), n=0..17); # Or:

%p hypergeom([2/3, 4/3], [1], 27*x): ser := series(%, x, 20):

%p seq(coeff(%, x, n), n=0..17); # Or:

%p a := proc(n) option remember; if n=0 then 1 else 3*(9 - n^(-2))*a(n-1) fi end:

%p # 4th Maple program:

%p W:=proc(x)sqrt(3)*MeijerG([[], [0, 0]], [[1/3, -1/3], []], x/27)/(18*Pi);end;

%p a:=proc(n) round(evalf[32](int(x^n*W(x),x=0..27)));end;

%p seq(a(n),n=0..17);

%p # _Karol A. Penson_, Jul 28 2023

%t Table[(3*n+1)*Binomial[3*n,n]*Binomial[2*n,n], {n,0,25}] (* _G. C. Greubel_, Mar 22 2022 *)

%o (Magma) [(n+1)^2*Binomial(3*n+1,n+1)*Catalan(n): n in [0..25]]; // _G. C. Greubel_, Mar 22 2022

%o (Sage) [(3*n+1)*binomial(2*n,n)*binomial(3*n,n) for n in (0..25)] # _G. C. Greubel_, Mar 22 2022

%Y Cf. A000108, A006480, A117671, A331431.

%K nonn

%O 0,2

%A _Peter Luschny_, Jan 18 2020