A086205 -id:A086205 - cp's OEIS frontend

A057863 a(n) = Product_{k=1..n} (2k-1)!!.

1, 1, 3, 45, 4725, 4465125, 46414974375, 6272287562165625, 12714083695698776015625, 438120013555654794702228515625, 286849911214281324754704976473779296875, 3943988517696329309474874414036059896739501953125

Offset: 0

Eric W. Weisstein

a(n) is the coefficient of the closed form for BarnesG[(2n-1)/2].
a(n) is the hook product corresponding to the partition (n,n-1,...,2,1). a(n)=(n(n+1)/2)!/A005118(n+1). - Emeric Deutsch, May 21 2004
Hankel transform of A185998. - Paul Barry, Feb 08 2011
The Burchnall-Chaundy polynomials P_n(z) have leading term z^(n^2+n)/a(n). - Michael Somos, Jan 18 2023

			G.f. = 1 + x + 3*x^2 + 45*x^3 + 4725*x^4 + 4465125*x^5 + ... - _Michael Somos_, Jan 18 2023

Alois P. Heinz, Table of n, a(n) for n = 0..35
Alejandro H. Morales, Igor Pak, and Greta Panova, Hook formulas for skew shapes III. Multivariate and product formulas, arXiv:1707.00931 [math.CO], 2017.
A. P. Veselov and R. Wilcox, Burchnall-Chaundy polynomials and the Laurent Phenomenon, arXiv:1407.7394 [math-ph], 2014.
Eric Weisstein's World of Mathematics, Barnes G-Function

Cf. A000178, A005118, A074962, A086205, A089626.

a:= n-> mul((2*k+1)^(n-k), k=0..n):
seq(a(n), n=0..15);  # Alois P. Heinz, Nov 28 2012

a[n_] := Product[2^i Gamma[1/2+i]/Sqrt[Pi], {i, n}]
Table[Product[(2*k+1)^(n-k),{k,0,n}],{n,0,10}] (* Vaclav Kotesovec, Nov 13 2014 *)
Table[Product[(2k-1)!!,{k,1,n}],{n,0,10}] (* Vaclav Kotesovec, Nov 13 2014 *)
Table[2^(n(n+1)/2-1/24) Glaisher^(3/2) Pi^(-n/2-1/4) E^(-1/8) BarnesG[n+3/2], {n, 0, 10}] (* Vladimir Reshetnikov, Nov 06 2015 *)
Table[Sqrt[BarnesG[2*n + 2]] / (2^(n^2/2) * BarnesG[n+1] * Sqrt[Gamma[n+1]]), {n, 0, 12}] (* Vaclav Kotesovec, Apr 08 2021 *)

PARI
```
a(n)=prod(k=0,n-1,prod(i=0,k,2*i+1))
```

a(n) = Product_{k=0..n} (2*k+1)^(n-k).
a(n) ~ A^(1/2) * 2^(n^2/2+n+5/24) * n^(n^2/2+n/2+1/24) / exp(3*n^2/4+n/2+1/24), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 13 2014
a(n) = 2^(n*(n+1)/2-1/24) * A^(3/2) * Pi^(-n/2-1/4) * exp(-1/8) * G(n+3/2), where A is the Glaisher-Kinkelin constant, G is the Barnes G-function. - Vladimir Reshetnikov, Nov 06 2015
a(n) = sqrt(G(2*n+2)) / (2^(n^2/2) * G(n+1) * sqrt(Gamma(n+1))), where G is the Barnes G-function. - Vaclav Kotesovec, Apr 08 2021
From Michael Somos, Jan 18 2023: (Start)
a(n) = (-1)^floor((n+1)/2)*a(-1-n) for all n in Z.
a(n+1)*a(n-1) = (2*n+1)*a(n)^2 for all n in Z.
(4*n + 8)*a(n+1)^2*a(n+2)^2 = a(n)*a(n+2)^3 + a(n+1)^3*a(n+3) for all n in Z.(End)
a(n) = (1/2^(n*(n-1)/2)) * A086205(n). - Peter Bala, Feb 20 2023

Simpler description from Benoit Cloitre, May 03 2003

Definition and programs corrected by Vaclav Kotesovec, Nov 13 2014

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)).

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)

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

A266091 a(n) = Product_{k=0..n} (3*k)!/(n+k)!.

Original entry on oeis.org

1, 3, 15, 126, 1782, 42471, 1706562, 115640460, 13216815036, 2548124192970, 828751754742975, 454739496669274500, 420972227408592675000, 657522745057190417409000, 1732789066323343611643088400, 7704900186426840030325195822560, 57807195523790513335568376591463776

Offset: 0

Michel Marcus, Dec 21 2015

a(n) gives the number of diagonally and antidiagonally symmetric alternating sign matrices (DASASM's) of order (2n+1) X (2n+1) (see Behrend et al. link).

Roger E. Behrend, Ilse Fischer, Matjaž Konvalinka, Diagonally and antidiagonally symmetric alternating sign matrices of odd order, arXiv:1512.06030 [math.CO], 2015.

Cf. A005157, A086205.

[&*[Factorial(3*k)/Factorial(n+k): k in [0..n]]: n in [0..16]]; // Vincenzo Librandi, Dec 21 2015

Table[Product[(3 k)!/(n + k)!, {k, 0, n}], {n, 0, 16}] (* Vincenzo Librandi, Dec 21 2015 *)

PARI
```
a(n) = prod(k=0, n, (3*k)!/(n+k)!);
```

a(n) ~ Gamma(1/3)^(1/3) * exp(1/36) * n^(1/36) * 3^(3*n^2/2 + 2*n + 11/36) / (A^(1/3) * Pi^(1/6) * 2^(2*n^2 + 2*n + 7/12)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Dec 21 2015

a(n) = Product_{1 <= i <= j <= n} (i + 2*j)/(i + j - 1). Note that Product_{1 <= i <= j <= n} (i + j)/(i + j - 1) = 2^n. - Peter Bala, Feb 19 2023

cp's OEIS Frontend

A057863 a(n) = Product_{k=1..n} (2k-1)!!.

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A268196 a(n) = Product_{k=0..n} binomial(3*k,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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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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A266091 a(n) = Product_{k=0..n} (3*k)!/(n+k)!.

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