A203471 -id:A203471 - 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

A203470 a(n) = Product_{2 <= i < j <= n+1} (i + j).

Original entry on oeis.org

1, 5, 210, 105840, 838252800, 129459762432000, 466521199899955200000, 45727437650097816797184000000, 139352822480378029387123167068160000000, 14863555768518278744824500982673408262144000000000, 61707340455179609358720715109663452970925870494515200000000000

Offset: 1

Clark Kimberling, Jan 02 2012

Each term divides its successor, as (conjectured) in A102693. Each term is divisible by the corresponding superfactorial, A000178(n), as in A203471.

G. C. Greubel, Table of n, a(n) for n = 1..36

Cf. A000178, A102693, A203471.

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

a:= n-> mul(mul(i+j, i=2..j-1), j=3..n+1):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017

(* First program *)
f[j_]:= j+1; 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}]           (* A203470 *)
Table[v[n+1]/v[n], {n, z-1}]  (* A102693 *)
Table[v[n]/d[n], {n, 20}]     (* A203471 *)
(* Second program *)
Table[Product[Gamma[2*j]/Gamma[j+2], {j,2,n+1}], {n,20}] (* G. C. Greubel, Aug 29 2023 *)

from math import prod, factorial
def A203470(n): return prod(factorial(2*k+1)//factorial(k+2) for k in range(1,n+1)) # Chai Wah Wu, Aug 26 2025

[product(gamma(2*k)/gamma(k+2) for k in range(2,n+2)) for n in range(1,20)] # G. C. Greubel, Aug 29 2023

a(n) ~ sqrt(A) * 2^(n^2 + 5*n/2 + 41/24) * exp(-3*n^2/4 + n/2 - 1/24) * n^(n^2/2 - n/2 - 71/24) / Pi, where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2021
From G. C. Greubel, Aug 29 2023: (Start)
a(n) = Product_{j=2..n+1} Gamma(2*j)/Gamma(j+2).
a(n) = (2/sqrt(Pi))*( 2^(n+1)^2 * BarnesG(n+5/2)/(Pi^(n/2)*Gamma(n+2)*Gamma(n+3)*BarnesG(3/2)) ).
a(n) = (BarnesG(n+2)/2^n) * Product_{j=2..n+1} Catalan(j). (End)

Name edited by Alois P. Heinz, Jul 23 2017

A296590 a(n) = Product_{k=0..n} binomial(2*n - k, k).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Dec 16 2017

Apart from the offset the same as A203469. - R. J. Mathar, Alois P. Heinz, Jan 02 2018

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

Cf. A001142, A007685, A086205, A098694, A110131, A112332, A203471, A268196, A296589, A296591, A338550.

A296590 := proc(n)
    mul( binomial(2*n-k,k),k=0..n) ;
end proc:
seq(A296590(n),n=0..7) ; # R. J. Mathar, Jan 03 2018

Table[Product[Binomial[2*n-k, k], {k, 0, n}], {n, 0, 15}]
Table[Glaisher^(3/2) * 2^(n^2 - 1/24) * BarnesG[n + 3/2] / (E^(1/8) * Pi^(n/2 + 1/4) * BarnesG[n + 2]), {n, 0, 15}]

a(n) = A^(3/2) * 2^(n^2 - 1/24) * BarnesG(n + 3/2) / (exp(1/8) * Pi^(n/2 + 1/4) * BarnesG(n + 2)), where A is the Glaisher-Kinkelin constant A074962.
a(n) ~ A^(3/2) * exp(n/2 - 1/8) * 2^(n^2 - 7/24) / (Pi^((n+1)/2) * n^(n/2 + 3/8)), where A is the Glaisher-Kinkelin constant A074962.
Product_{1 <= j <= i <= n}  (i + j - 1)/(i - j + 1). - Peter Bala, Oct 25 2024

A338550 Number of binary trees of height n such that the number of nodes at depth d equals d+1 for every d = 0..n.

Original entry on oeis.org

1, 1, 4, 60, 3360, 705600, 558835200, 1678182105600, 19198403288064000, 840083731079104512000, 141100463472046393835520000, 91242050302344912388163665920000, 227753296409896438988240405704212480000, 2199573010737856838816729366169572868096000000, 82356764599728553816070191604819734458909327360000000

Offset: 0

Marcel K. Goh, Nov 02 2020

To satisfy the constraint, there must be n+1 nodes at depth n, and there are 2n allowed slots for a new node.

A binary tree with such a level profile contains A000217(n+1) nodes.

Cf. A000217, A007685, A203471, A296589.

Table[Product[Binomial[2k,k+1],{k,n}],{n,0,14}] (* or *)
Table[2^(n^2+n-1/24)Glaisher^(3/2)Pi^(-1/4-n/2)BarnesG[3/2+n]Gamma[1+n]/(Exp[1/8]BarnesG[3+n]),{n,0,14}] (* Stefano Spezia, Nov 02 2020 *)

a(n) = binomial(2*n,n+1)*a(n-1), a(0)=1.
a(n) = Product_{k=1..n} binomial(2*k,k+1).
a(n) = 2^(n^2+n-1/24)*A^(3/2)*Pi^(-1/4-n/2)*G(3/2 + n)*Gamma(1 + n)/(exp(1/8)*G(3 + n)) where A is the Glaisher-Kinkelin constant and G is the Barnes G function. - Stefano Spezia, Nov 02 2020
a(n) ~ A^(3/2) * 2^(-7/24 + n + n^2) * exp(-1/8 + n/2) / (n^(11/8 + n/2) * Pi^((n+1)/2)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 29 2023
a(n) = Product_{1 <= j <= i <= n-1}  (i + j + 2)/(i - j + 1). - Peter Bala, Oct 25 2024

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

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A203470 a(n) = Product_{2 <= i < j <= n+1} (i + j).

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A296590 a(n) = Product_{k=0..n} binomial(2*n - k, k).

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A338550 Number of binary trees of height n such that the number of nodes at depth d equals d+1 for every d = 0..n.

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