A112332 -id:A112332 - 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

A268196 a(n) = Product_{k=0..n} binomial(3*k,k).

Original entry on oeis.org

1, 3, 45, 3780, 1871100, 5618913300, 104309506501200, 12129109415959536000, 8920608231265175901456000, 41809329673499408044341517200000, 1256161937180234817183361549396758000000, 243113461110708695347467432844366521953760000000

Offset: 0

Vaclav Kotesovec, Apr 16 2016

Harvey P. Dale, Table of n, a(n) for n = 0..49
Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function.

Cf. A001142, A005809, A007685, A086205, A112332, A268504, A262261.

Table[Product[Binomial[3k,k],{k,0,n}],{n,0,12}]
FoldList[Times,Table[Binomial[3n,n],{n,0,15}]] (* Harvey P. Dale, Apr 23 2018 *)

a(n) = A^(7/6) * Gamma(1/3)^(1/3) * 3^(3*n^2/2 + 2*n + 11/36)* BarnesG(n + 4/3) * BarnesG(n + 5/3) / (exp(7/72) * 2^(n^2 + 2*n + 5/8) * Pi^(n/2 + 5/12) * BarnesG(n + 3/2) * BarnesG(n + 2)), where A = A074962 is the Glaisher-Kinkelin constant.
a(n) ~ A^(7/6) * Gamma(1/3)^(1/3) * 3^(11/36 + 2*n + 3*n^2/2) * exp(n/2 - 7/72) / (2^(n^2 + 2*n + 7/8) * Pi^(n/2 + 2/3) * n^(n/2 + 25/72)), where A = A074962 is the Glaisher-Kinkelin constant.
a(n) = A268504(n) / (A000178(n) * A098694(n)).

A306594 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i + j + k).

Original entry on oeis.org

1, 3, 144000, 455282248974336000000, 9608917807566747651759509633033255126040576000000000000

Offset: 0

Vaclav Kotesovec, Feb 27 2019

nonn

Next term is too long to be included.

Cf. A079478, A093884, A112332, A324425, A368722, A368723, A324441.

a:= n-> mul(mul(mul(i+j+k, i=1..n), j=1..n), k=1..n):
seq(a(n), n=0..5);  # Alois P. Heinz, Jun 24 2023

Table[Product[i+j+k, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 1, 6}]
Table[Product[k^(3*(n - k + 1) (n - k + 2)/2), {k, 1, n}] * Product[k^((3*n - k + 1) (3*n - k + 2)/2), {k, 1, 3*n}] / Product[k^(3*(2*n - k + 1) (2*n - k + 2)/2), {k, 1, 2*n}], {n, 1, 6}]
Clear[a]; a[n_] := a[n] = If[n == 1, 3, 3*n*a[n-1] * BarnesG[2+n]^3 * BarnesG[2+3*n]^3 * Gamma[1+2*n]^3 / (BarnesG[2+2*n]^6 * Gamma[1+3*n]^3)]; Table[a[n], {n, 1, 6}] (* Vaclav Kotesovec, Mar 28 2019 *)

a(n) = Product_{k=1..n} (BarnesG(k+2) * BarnesG(2*n+k+2) / BarnesG(n+k+2)^2).
a(n) = Product_{k=1..n} (k^(3*(n - k + 1)*(n - k + 2)/2)) * Product_{k=1..3*n} (k^((3*n - k + 1)*(3*n - k + 2)/2)) / Product_{k=1..2*n} (k^(3*(2*n - k + 1)*(2*n - k + 2)/2)).
a(n) ~ sqrt(Pi) * 3^(9*n^3/2 + 27*n^2/4 + 3*n + 3/8) * n^(n^3 + 3/8) / (A^(3/2) * 2^(4*n^3 + 9*n^2 + 6*n + 5/8) * exp(11*n^3/6 - Zeta(3)/(8*Pi^2) - 1/8)), where A is the Glaisher-Kinkelin constant A074962.

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

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

A110131 Determinant of n X n matrix M_{i,j} = 2^i*P_i(j), where P_i(j) is the Legendre polynomial of order i at j and i and j are 0-based.

Original entry on oeis.org

1, 2, 24, 2880, 4838400, 146313216000, 97339256340480000, 1683704371913057894400000, 873705178746128941669416960000000, 15414977576506278044562764045746176000000000, 10334857226047177887548812577909403133201612800000000000

Offset: 1

Paul Barry, Jul 13 2005

Muniru A Asiru, Table of n, a(n) for n = 1..36

Cf. A000079, A086205, A112332.

seq(mul(mul((j+k),j=1..k), k=1..n-1), n=1..9); # Zerinvary Lajos, Sep 21 2007

a(n)=my(t=1);prod(k=1,n-1,t*=4*k-2) \\ Charles R Greathouse IV, Oct 25 2011

a(n)=matdet(matrix(n,n,i,j,pollegendre(i-1,j-1)<<(i-1))) \\ Charles R Greathouse IV, Oct 25 2011

a(n) = 2^n * Product_{k=1..n} (2*k-1)!/(k-1)!.
a(n) = 2^n * A086205(n).
From Alois P. Heinz, Jun 30 2022: (Start)
a(n) = Product_{i=1..n-1} Product_{j=i..n-1} (i+j).
a(n) = A112332(n). (End)

A355400 Number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths of semilength n, such that each p_i is never below p_{i-1}.

Original entry on oeis.org

1, 1, 3, 30, 1001, 111384, 41314284, 51067020290, 210309203300625, 2885318087540733000, 131857099297936066411200, 20070377346929658409924542720, 10174783866874800701945612292557712, 17178820188393063395267380511228827387600, 96592800670609299321035523895170598736583965100

Offset: 0

Alois P. Heinz, Jun 30 2022

nonn

Determinant of the n X n Hankel matrix whose i-th antidiagonal is filled with the n+i-th Catalan number for i = 0..2*n-2.
              [ 5,  14,  42]
  a(3) = det( [14,  42, 132] ) = 30.
              [42, 132, 429]

			a(0) = 1:  ( ).
a(1) = 1:  (/\).
a(2) = 3:                        /\      /\    /\
           (/\/\, /\/\), (/\/\, /  \), (/  \, /  \).
G.f. = 1 + x + 3*x^2 + 30*x^3 + 1001*x^4 + 111384*x^5 + 41314284*x^6 + ... - _Michael Somos_, Jun 27 2023

Alois P. Heinz, Table of n, a(n) for n = 0..62
Wikipedia, Counting lattice paths
Wikipedia, Hankel matrix

A diagonal of A078920 or of A123352 or of A368025.

Cf. A000108, A006125, A074962, A076113, A110131, A112332, A131577, A358597, A368298, A378113.

a:= n-> mul(mul((i+j+2*n)/(i+j), j=i..n-1), i=1..n-1):
seq(a(n), n=0..14);

Join[{1}, Table[Sqrt[2*BarnesG[4*n]] * BarnesG[n] * Gamma[2*n]^(3/2) / BarnesG[3*n + 1], {n, 1, 12}]] (* Vaclav Kotesovec, Aug 26 2023 *)

a(n) = prod(i=1, n-1, prod(j=i, n-1, (i+j+2*n)/(i+j))); \\ Michel Marcus, Jul 05 2022

a(n) = Product_{i=1..n-1, j=i..n-1} (i+j+2*n)/(i+j).
a(n) mod 2 = 1 <=> n in { A131577 }.
a(n) ~ exp(1/24) * 2^(1/6 - n + 8*n^2) / (sqrt(A) * n^(1/24) * 3^(9*n^2/2 - 1/12)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 26 2023

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

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

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

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, May 14 2005

nonn

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

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

[1] cat [(&*[Factorial(n+k)/Factorial(k+1): k in [0..n-1]]): n in [1..10]]; // G. C. Greubel, May 21 2019

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 *)

{a(n) = prod(k=0,n-1, (n+k)!/(k+1)!)}; \\ G. C. Greubel, May 21 2019

[product(factorial(n+k)/factorial(k+1) for k in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 21 2019

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

A324427 a(n) = Product_{k=1..n} (Product_{j=1..k} (Product_{i=1..j} (i+j+k))).

Original entry on oeis.org

1, 3, 360, 38102400, 109506663383040000, 337878174593229551661219840000000, 54048023654871725380569225530796717972337459200000000000, 25571582464158460440549345359703385621119611033206432205259362823202406400000000000000000

Offset: 0

Vaclav Kotesovec, Feb 27 2019

nonn

Cf. A079478, A112332, A306594.

a:= n-> mul(mul(mul(i+j+k, i=1..j), j=1..k), k=1..n):
seq(a(n), n=0..8);  # Alois P. Heinz, Jun 24 2023

Table[Product[Product[Product[i+j+k, {i, 1, j}], {j, 1, k}], {k, 1, n}], {n, 0, 10}]
Table[Sqrt[Product[2^k Gamma[1 + 3*k/2]/Gamma[1 + k/2] (BarnesG[2 + k] BarnesG[2 + 3 k] )/BarnesG[2 + 2 k]^2 , {k, 1, n}]], {n, 0, 10}]

a(n) = prod(k=1, n, prod(j=1, k, prod(i=1, j, i+j+k))); \\ Michel Marcus, Feb 27 2019

a(n) ~ 3^(3*n^3/4 + 9*n^2/4 + 47*n/24 + 7/24) * n^(n^3/6 + n^2/2 + n/3) / (2^(2*n^3/3 + 2*n^2 + 7*n/4 + 7/24) * exp(11*n^3/36 + 3*n^2/4 + n/3 - zeta(3)/(48*Pi^2))). - Vaclav Kotesovec, Nov 27 2023

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

cp's OEIS Frontend

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

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A268196 a(n) = Product_{k=0..n} binomial(3*k,k).

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A306594 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i + j + k).

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

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A110131 Determinant of n X n matrix M_{i,j} = 2^i*P_i(j), where P_i(j) is the Legendre polynomial of order i at j and i and j are 0-based.

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A355400 Number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths of semilength n, such that each p_i is never below p_{i-1}.

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

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

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