A296590 -id:A296590 - cp's OEIS frontend

A007685 a(n) = Product_{k=1..n} binomial(2*k,k).

1, 2, 12, 240, 16800, 4233600, 3911846400, 13425456844800, 172785629592576000, 8400837310791045120000, 1552105098192510332190720000, 1094904603628138948657963991040000, 2960792853328653706847125274154762240000, 30794022150329995743434211126374020153344000000

Offset: 0

N. J. A. Sloane

H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..50
H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)
Bernd C. Kellner, Asymptotic products of binomial and multinomial coefficients revisited, Integers 24 (2024), Article #A59, 10 pp.; arXiv:2312.11369 [math.CO], 2023.

Cf. A000984, A001142, A112332, A203471, A268196, A296589, A296590, A338550.

[seq(mul(binomial(2*k,k),k=1..n),n=0..16)];

Table[Product[Binomial[2*k, k], {k, 1, n}], {n, 0, 50}] (* G. C. Greubel, Feb 02 2017 *)

a(n) = prod(k=1,n, binomial(2*k, k)); \\ Michel Marcus, Sep 18 2015

a(0) = 1, a(n) = (2^(2*n)*a(n - 1)*Gamma(n + 1/2))/(sqrt(Pi)*Gamma(n + 1)). - Ilya Gutkovskiy, Sep 18 2015
a(n) = (2^(n^2 + n - 1/24)*A^(3/2)*Pi^(-n/2 - 1/4)*BarnesG(n + 3/2))/(e^(1/8)*BarnesG(n + 2)), where A is the Glaisher-Kinkelin constant (A074962), BarnesG is the Barnes G-function. - Ilya Gutkovskiy, Sep 18 2015
a(n) ~ A^(3/2) * 2^(n^2 + n - 7/24) * exp(n/2 - 1/8) / (Pi^((n+1)/2) * n^(n/2 + 3/8)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Apr 16 2016
For n>0, a(n) = 2^((n+1)/2) * sqrt(BarnesG(2*n)) * Gamma(2*n) / (n * BarnesG(n)^2 * Gamma(n)^(7/2)). - Vaclav Kotesovec, Apr 20 2024
Product_{1 <= j <= i <= n}  (i + j)/(i - j + 1). - Peter Bala, Oct 25 2024

A296591 a(n) = Product_{k=0..n} (n + k)!.

Original entry on oeis.org

1, 2, 288, 12441600, 421382062080000, 23120161750363668480000000, 3683853104727992382799761899520000000000, 2777528195026874073410445622205453260145295360000000000000

Offset: 0

Vaclav Kotesovec, Dec 16 2017

nonn

Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function

Cf. A001142, A007685, A086205, A098694, A110131, A112332, A268196, A296589, A296590.

a:= proc(n) option remember; `if`(n=0, 1,
      a(n-1) *(2*n-1)! *(2*n)! /(n-1)!)
    end:
seq(a(n), n=0..7);  # Alois P. Heinz, Jul 11 2024

Table[Product[(n + k)!, {k, 0, n}], {n, 0, 10}]
Table[Product[(2*n - k)!, {k, 0, n}], {n, 0, 10}]
Table[BarnesG[2*n + 2]/BarnesG[n + 1], {n, 0, 10}]

a(n) = BarnesG(2*n + 2) / BarnesG(n + 1).

a(n) ~ 2^(2*n^2 + 5*n/2 + 11/12) * n^((n+1)*(3*n+1)/2) * Pi^((n+1)/2) / exp(9*n^2/4 + 2*n).

Missing a(0)=1 inserted by Georg Fischer, Nov 18 2021

A296589 a(n) = Product_{k=0..n} binomial(2*n, k).

Original entry on oeis.org

1, 2, 24, 1800, 878080, 2857680000, 63117561830400, 9577928124440387712, 10077943267571584204800000, 74054886893191804566576837427200, 3822038592032831128918160803430400000000, 1391938996758770867922655936144556115037409280000

Offset: 0

Vaclav Kotesovec, Dec 16 2017

nonn

Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function

Cf. A001142, A007685, A086205, A110131, A112332, A268196, A296590, A296591.

Table[Product[Binomial[2*n, k], {k, 0, n}], {n, 0, 12}]
Table[((2*n)!)^(n+1) / (n! * BarnesG[2*n + 2]), {n, 0, 12}]

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

a(n) ~ A * exp(n^2 + n - 1/24) / (2^(5/12) * Pi^((n+1)/2) * n^(n/2 + 5/12)), where A is the Glaisher-Kinkelin constant A074962.

Missing a(0)=1 inserted by Georg Fischer, Nov 18 2021

A203469 a(n) = v(n)/A000178(n), v = A093883 and A000178 = (superfactorials).

Original entry on oeis.org

1, 3, 30, 1050, 132300, 61122600, 104886381600, 674943865596000, 16407885372638760000, 1515727634953623371280000, 534621388490302221024396480000, 722849817707190846398223943885440000, 3759035907022704558524683975387453632000000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..55

Cf. A000178, A006963, A076756, A093883, A296590.

[(&*[Binomial(2*n-k,k): k in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 29 2023

(* First program *)
f[j_]:= j; z = 16;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,z}]           (* A093883 *)
Table[v[n+1]/v[n], {n,z-1}]  (* A006963 *)
Table[v[n]/d[n], {n,20}]     (* A203469 *)
(* Second program *)
Table[Product[Binomial[2*n-j,j], {j,n}], {n,20}] (* G. C. Greubel, Aug 29 2023 *)

[product(binomial(2*n-j,j) for j in range(n)) for n in range(1, 20)] # G. C. Greubel, Aug 29 2023

a(n) = Product_{i=1..n} binomial(2n-i,i).  - Enrique Pérez Herrero, Feb 20 2013
From G. C. Greubel, Aug 29 2023: (Start)
a(n) = (2^n/sqrt(Pi))^n*BarnesG(n+3/2)/(BarnesG(n+2)*BarnesG(3/2)).
a(n) = (n!/2^(n-1))*Product_{j=1..n-1} Catalan(j). (End)
a(n) ~ A^(3/2) * exp(n/2 - 1/8) * 2^(n^2 - 7/24) / (Pi^(n/2 + 1/2) * n^(n/2 + 3/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 26 2023

cp's OEIS Frontend

A007685 a(n) = Product_{k=1..n} binomial(2*k,k).

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A296591 a(n) = Product_{k=0..n} (n + k)!.

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A296589 a(n) = Product_{k=0..n} binomial(2*n, k).

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Mathematica

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A203469 a(n) = v(n)/A000178(n), v = A093883 and A000178 = (superfactorials).

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