login
a(n) = Product_{k=0..n} C(n+k+2, n+2).
2

%I #12 Aug 29 2023 09:59:57

%S 1,4,75,7056,3457440,9032601600,127843321480875,9917120529316000000,

%T 4253520573615071657074176,10156681309872614660803421798400,

%U 135766978921156343322148046967386880000,10205737152660536205131284348877857357824000000

%N a(n) = Product_{k=0..n} C(n+k+2, n+2).

%C Divisors in number triangle A120247.

%H G. C. Greubel, <a href="/A120248/b120248.txt">Table of n, a(n) for n = 0..50</a>

%F a(n) = Gamma(n+2)*BarnesG(2*n+4)/((Gamma(n+3))^(n-1)*BarnesG(n+4)^2). - _G. C. Greubel_, Mar 16 2023

%F a(n) ~ A * 2^(47/12 + 11*n/2 + 2*n^2) / (exp(19/6 + 2*n + n^2/2) * Pi^((n+1)/2) * n^(5/12 + n/2)), where A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Aug 29 2023

%t Table[Gamma[n+2]*BarnesG[2*n+4]/((Gamma[n+3])^(n-1)*BarnesG[n+4]^2), {n,0,20}] (* _G. C. Greubel_, Mar 16 2023 *)

%o (Magma)

%o A120248:= func< n | (&*[Binomial(n+j+2, n+2): j in [0..n]]) >;

%o [A120248(n): n in [0..20]]; // _G. C. Greubel_, Mar 16 2023

%o (SageMath)

%o def A120248(n): return product( binomial(n+j+2, n+2) for j in range(n+1))

%o [A120248(n) for n in range(21)] # _G. C. Greubel_, Mar 16 2023

%o (PARI) a(n) = prod(k=0, n, binomial(n+k+2, n+2)); \\ _Michel Marcus_, Mar 16 2023

%Y Cf. A120247.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jun 12 2006