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 1-5 of 5 results.

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

Original entry on oeis.org

1, 2, 12, 240, 16800, 4233600, 3911846400, 13425456844800, 172785629592576000, 8400837310791045120000, 1552105098192510332190720000, 1094904603628138948657963991040000, 2960792853328653706847125274154762240000, 30794022150329995743434211126374020153344000000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • 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).

Links

Crossrefs

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

Programs

  • Maple
    [seq(mul(binomial(2*k,k),k=1..n),n=0..16)];
  • Mathematica
    Table[Product[Binomial[2*k, k], {k, 1, n}], {n, 0, 50}] (* G. C. Greubel, Feb 02 2017 *)
  • PARI
    a(n) = prod(k=1,n, binomial(2*k, k)); \\ Michel Marcus, Sep 18 2015

Formula

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

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

A203471 a(n) = v(n)/A000178(n), v = A203470, A000178 = (superfactorials).

Original entry on oeis.org

1, 5, 105, 8820, 2910600, 3745942200, 18748440711000, 364619674947528000, 27558684271884061296000, 8100324068034882136733280000, 9267305355220395466643896716480000, 41308086890359390753018505224037952000000, 718093999369842998535768272223944392254720000000
Offset: 1

Views

Author

Clark Kimberling, Jan 02 2012

Keywords

Links

Crossrefs

Cf. A000178, A007685, A102693, A203470, A296589, A338550.

Programs

  • Magma
    [(&*[Factorial(2*k+1)/(Factorial(k-1)*Factorial(k+2)): k in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 29 2023
  • Mathematica
    (* 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+2]/(Gamma[j]*Gamma[j+3]), {j,n}], {n,20}] (* G. C. Greubel, Aug 29 2023 *)
  • SageMath
    [product(gamma(2*k+4)/(gamma(k+1)*gamma(k+4)) for k in range(n)) for n in range(1, 20)] # G. C. Greubel, Aug 29 2023

Formula

From G. C. Greubel, Aug 29 2023: (Start)
a(n) = Product_{j=1..n} Gamma(2*j+2)/(Gamma(j)*Gamma(j+3)).
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)*BarnesG(n+1)) ).
a(n) = (BarnesG(n+2)/(2^n * BarnesG(n+1))) * Product_{j=1..n} Catalan(j+1). (End)
a(n) ~ A^(3/2) * 2^(n^2 + 2*n + 41/24) * exp(n/2 - 1/8) / (n^(n/2 + 23/8) * Pi^(n/2 + 1)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 19 2023
a(n) = Product_{1 <= j <= i <= n-1} (i + j + 3)/(i - j + 1). - Peter Bala, Oct 25 2024

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

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

Views

Author

Marcel K. Goh, Nov 02 2020

Keywords

Comments

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.

Crossrefs

Cf. A000217, A007685, A203471, A296589.

Programs

  • Mathematica
    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 *)

Formula

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
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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