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A107252 a(n) = Product_{k=0..n-1} (n+k)!/(k+1)!.

%I A107252 #21 Sep 08 2022 08:45:18
%S A107252 1,1,6,1440,36288000,184354652160000,309071606732292096000000,
%T A107252 254046582743105184577722777600000000,
%U A107252 142518177743863255019484504453155074867200000000000
%N A107252 a(n) = Product_{k=0..n-1} (n+k)!/(k+1)!.
%F A107252 a(n) = (n+1)!*(n+2)!*...*(2n-1)!/(1!*2!*...*(n-1)!).
%F A107252 a(n) = A000178(2n-1)/(A000178(n)*A000178(n-1)).
%F A107252 a(n) = A079478(n)/A001813(n).
%F A107252 a(n) = A079478(n-1)*A006963(n+1).
%F A107252 a(n) = A107251(n)/A000108(n).
%F A107252 a(n) = A107251(n-1)*A009445(n-1).
%F A107252 a(n) = A107254(n)/A000142(n).
%F A107252 a(n) = A009963(2n-1, n-1).
%F A107252 a(n) = A009963(2n-1, n).
%F A107252 a(n) = (G(1+2*n)*n!*((G(2+n)*Gamma(2+n))/G(3+n))^(n-1))/G(2+n)^2, where G(x) is the Barnes G function. - _Peter Luschny_, May 20 2019
%F A107252 a(n) ~ A * 2^(2*n^2 - 7/12) * n^(n^2 - n - 5/12) / (sqrt(Pi) * exp(3*n^2/2 - n + 1/12)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, May 21 2019
%e A107252 a(3) = 1!*2!*3!*4!*5!/(1!*2!*3!*1!*2!) = 34560/24 = 1440.
%t A107252 Table[Product[(n+k)!/(k+1)!,{k,0,n-1}],{n,0,10}] (* _Alexander Adamchuk_, Jul 10 2006 *)
%t A107252 a[n_] := (BarnesG[1 + 2 n] n! ((BarnesG[2 + n] Gamma[2 + n])/ BarnesG[3 + n])^(-1 + n))/BarnesG[2 + n]^2; Table[a[n], {n, 0, 10}] (* _Peter Luschny_, May 20 2019 *)
%o A107252 (PARI) {a(n) = prod(k=0,n-1, (n+k)!/(k+1)!)}; \\ _G. C. Greubel_, May 21 2019
%o A107252 (Magma) [1] cat [(&*[Factorial(n+k)/Factorial(k+1): k in [0..n-1]]): n in [1..10]]; // _G. C. Greubel_, May 21 2019
%o A107252 (Sage) [product(factorial(n+k)/factorial(k+1) for k in (0..n-1)) for n in (0..10)] # _G. C. Greubel_, May 21 2019
%Y A107252 Cf. A000178, A001700, A005130, A009963, A110131, A112332, A107254, A000142.
%K A107252 nonn
%O A107252 0,3
%A A107252 _Henry Bottomley_, May 14 2005