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

1, 1, 6, 1440, 36288000, 184354652160000, 309071606732292096000000, 254046582743105184577722777600000000, 142518177743863255019484504453155074867200000000000

Offset

0,3

Links

Table of n, a(n) for n=0..8.

Formula

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

a(n) = A000178(2n-1)/(A000178(n)*A000178(n-1)).

a(n) = A079478(n)/A001813(n).

a(n) = A079478(n-1)*A006963(n+1).

a(n) = A107251(n)/A000108(n).

a(n) = A107251(n-1)*A009445(n-1).

a(n) = A107254(n)/A000142(n).

a(n) = A009963(2n-1, n-1).

a(n) = A009963(2n-1, n).

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

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

Example

a(3) = 1!*2!*3!*4!*5!/(1!*2!*3!*1!*2!) = 34560/24 = 1440.

Mathematica

Table[Product[(n+k)!/(k+1)!, {k, 0, n-1}], {n, 0, 10}] (* Alexander Adamchuk, Jul 10 2006 *)
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 *)

Prog

(PARI) {a(n) = prod(k=0, n-1, (n+k)!/(k+1)!)}; \\ G. C. Greubel, May 21 2019
(Magma) [1] cat [(&*[Factorial(n+k)/Factorial(k+1): k in [0..n-1]]): n in [1..10]]; // G. C. Greubel, May 21 2019
(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

Crossrefs

Cf. A000178, A001700, A005130, A009963, A110131, A112332, A107254, A000142.

Sequence in context: A265473 A199645 A300025 * A358626 A131453 A265862

Adjacent sequences: A107249 A107250 A107251 * A107253 A107254 A107255

Keyword

nonn

Author

Henry Bottomley, May 14 2005

Status

approved