cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A112332 a(n) = Product_{k=0..n-1} k!*binomial(2k,k).

Original entry on oeis.org

1, 1, 2, 24, 2880, 4838400, 146313216000, 97339256340480000, 1683704371913057894400000, 873705178746128941669416960000000, 15414977576506278044562764045746176000000000, 10334857226047177887548812577909403133201612800000000000
Offset: 0

Views

Author

Paul Barry, Sep 04 2005

Keywords

Crossrefs

Cf. A005130, A110131.

Programs

  • Maple
    seq(mul(mul((j+k),j=1..k), k=1..n), n=-1..9); # Zerinvary Lajos, Sep 21 2007
  • Mathematica
    Table[Product[(2*k)!/k!,{k,0,n-1}],{n,0,10}] (* Vaclav Kotesovec, Jul 11 2015 *)

Formula

a(n)=denominator(Product{k=0..n-1, (2k+1)!/(n+k)!}).
G.f.: 1+ x*G(0)/2, where G(k)= 1 + 1/(1 - 1/(1 + 1/((2*k+2)!/(k+1)!)/x/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013
a(n) ~ A^(1/2) * 2^(n^2 - n/2 - 7/24) * n^(n^2/2 - n/2 + 1/24) / exp(3*n^2/4 - n/2 + 1/24), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 11 2015
From Alois P. Heinz, Jun 30 2022: (Start)
a(n) = Product_{i=1..n-1} Product_{j=i..n-1} (i+j).
a(n) = A110131(n). (End)

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

Original entry on oeis.org

1, 2, 288, 12441600, 421382062080000, 23120161750363668480000000, 3683853104727992382799761899520000000000, 2777528195026874073410445622205453260145295360000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 16 2017

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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}]

Formula

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).

Extensions

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

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

Views

Author

Alois P. Heinz, Jun 30 2022

Keywords

Comments

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]

Examples

			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

Links

Crossrefs

A diagonal of A078920 or of A123352 or of A368025.
Cf. A000108, A006125, A074962, A076113, A110131, A112332, A131577, A358597, A368298, A378113.

Programs

  • Maple
    a:= n-> mul(mul((i+j+2*n)/(i+j), j=i..n-1), i=1..n-1):
seq(a(n), n=0..14);
  • Mathematica
    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 *)
  • PARI
    a(n) = prod(i=1, n-1, prod(j=i, n-1, (i+j+2*n)/(i+j))); \\ Michel Marcus, Jul 05 2022

Formula

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

Views

Author

Vaclav Kotesovec, Dec 16 2017

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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.

Extensions

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

Views

Author

Vaclav Kotesovec, Dec 16 2017

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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}]

Formula

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

Views

Author

Henry Bottomley, May 14 2005

Keywords

Examples

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

Crossrefs

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

Programs

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

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
Showing 1-6 of 6 results.

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