This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A331322 #44 Apr 23 2025 16:20:02
%S A331322 1,24,630,16800,450450,12108096,325909584,8779605120,236637794250,
%T A331322 6380456082000,172080900531540,4641917845743360,125235075213284400,
%U A331322 3379123922914656000,91184624634161304000,2460769070127233057280,66411927755894739034170,1792432652235221330334000
%N A331322 a(n) = (3*n + 1)!/(n!)^3.
%C A331322 Diagonal of the rational function 1 / (1 - x - y - z)^2. - _Ilya Gutkovskiy_, Apr 23 2025
%H A331322 G. C. Greubel, <a href="/A331322/b331322.txt">Table of n, a(n) for n = 0..500</a>
%F A331322 a(n) = [x^n] hypergeom([2/3, 4/3], [1], 27*x).
%F A331322 a(n) = 3*(9 - n^(-2))*a(n-1) for n > 0.
%F A331322 a(n) = (-1)^n*A331431(2*n, n).
%F A331322 a(n) = (n+1)^2*A117671(n)*A000108(n). - _G. C. Greubel_, Mar 22 2022
%F A331322 From _Karol A. Penson_, Jul 28 2023: (Start)
%F A331322 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 A331322 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 A331322 From _Peter Bala_, Oct 10 2024: (Start)
%F A331322 a(n) = (3*n + 1)*A006480(n).
%F A331322 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 A331322 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 A331322 a := n -> (3*n+1)!/(n!)^3: seq(a(n), n=0..17); # Or:
%p A331322 hypergeom([2/3, 4/3], [1], 27*x): ser := series(%, x, 20):
%p A331322 seq(coeff(%, x, n), n=0..17); # Or:
%p A331322 a := proc(n) option remember; if n=0 then 1 else 3*(9 - n^(-2))*a(n-1) fi end:
%p A331322 # 4th Maple program:
%p A331322 W:=proc(x)sqrt(3)*MeijerG([[], [0, 0]], [[1/3, -1/3], []], x/27)/(18*Pi);end;
%p A331322 a:=proc(n) round(evalf[32](int(x^n*W(x),x=0..27)));end;
%p A331322 seq(a(n),n=0..17);
%p A331322 # _Karol A. Penson_, Jul 28 2023
%t A331322 Table[(3*n+1)*Binomial[3*n,n]*Binomial[2*n,n], {n,0,25}] (* _G. C. Greubel_, Mar 22 2022 *)
%o A331322 (Magma) [(n+1)^2*Binomial(3*n+1,n+1)*Catalan(n): n in [0..25]]; // _G. C. Greubel_, Mar 22 2022
%o A331322 (Sage) [(3*n+1)*binomial(2*n,n)*binomial(3*n,n) for n in (0..25)] # _G. C. Greubel_, Mar 22 2022
%Y A331322 Cf. A000108, A006480, A117671, A331431.
%K A331322 nonn
%O A331322 0,2
%A A331322 _Peter Luschny_, Jan 18 2020