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 1-3 of 3 results.

A068553 a(n) = lcm(1,2,...,2*n) / (n*binomial(2*n, n)).

Original entry on oeis.org

1, 1, 1, 3, 2, 5, 15, 7, 28, 126, 30, 165, 198, 143, 1001, 15015, 3640, 884, 7956, 1938, 19380, 203490, 49742, 572033, 980628, 240350, 3124550, 766935, 188370, 2731365, 40970475, 20160075, 4962480, 81880920, 20173560, 353037300
Offset: 1

Views

Author

N. J. A. Sloane, Mar 23 2002

Keywords

Comments

Known to be always an integer.

Links

Crossrefs

Bisection of A048619.
Cf. A068550.

Programs

  • Magma
    [Lcm([1..2*n])/(n*(n+1)*Catalan(n)): n in [1..50]]; // G. C. Greubel, May 04 2023
  • Maple
    Num:= 2: Den:=2: Res:= 1:
for n from 2 to 100 do
  Num:= ilcm(Num,2*n-1,2*n);
  Den:= Den*(4+2/(n-1));
  Res:= Res, Num/Den;
od:
Res; # Robert Israel, Dec 26 2018
  • Mathematica
    Table[(LCM@@Range[2n])/(n Binomial[2n,n]),{n,40}] (* Harvey P. Dale, Jul 17 2012 *)
  • SageMath
    def A068553(n) -> int:
    return lcm(range(1,2*n+1))//(n*binomial(2*n,n))
[A068553(n) for n in range(1,51)] # G. C. Greubel, May 04 2023

Formula

a(n) = A068550(n)/n.
a(n) = A048619(2*n-1).

A068550 a(n) = lcm{1, ..., 2n} / binomial(2n, n).

Original entry on oeis.org

1, 1, 2, 3, 12, 10, 30, 105, 56, 252, 1260, 330, 1980, 2574, 2002, 15015, 240240, 61880, 15912, 151164, 38760, 406980, 4476780, 1144066, 13728792, 24515700, 6249100, 84362850, 21474180, 5462730, 81940950, 1270084725, 645122400
Offset: 0

Views

Author

N. J. A. Sloane, Mar 23 2002

Keywords

Comments

Known to be always an integer.

Links

Crossrefs

Bisection of A180000 and A263673.

Programs

  • Mathematica
    a[0] = 1; a[n_] := (LCM @@ Range[2*n])/Binomial[2*n, n]; Array[a, 33, 0] (* Amiram Eldar, Mar 06 2022 *)
  • PARI
    a(n) = lcm([1..2*n])/binomial(2*n, n); \\ Michel Marcus, Mar 06 2022

Formula

a(n) = A099996(n) / A000984(n) = A003418(2*n) / A001405(2*n) = A180000(2*n) = A263673(2*n).
a(n) = n * A068553(n) = n * A048619(2*n-1).

Extensions

a(0)=1 prepended by Max Alekseyev, Oct 23 2015

A263673 a(n) = lcm{1,2,...,n} / binomial(n,floor(n/2)).

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 3, 12, 12, 20, 10, 60, 30, 210, 105, 56, 56, 504, 252, 2520, 1260, 660, 330, 3960, 1980, 5148, 2574, 4004, 2002, 30030, 15015, 240240, 240240, 123760, 61880, 31824, 15912, 302328, 151164, 77520, 38760, 813960, 406980, 8953560, 4476780, 2288132, 1144066, 27457584, 13728792, 49031400
Offset: 0

Views

Author

Max Alekseyev, Oct 23 2015

Keywords

Comments

From Robert Israel, Oct 23 2015: (Start)
If n = 2^k, a(n) = a(n-1).
If n = p^k where p is an odd prime and k >= 1, 2*n*a(n) = p*(n+1)*a(n-1).
If n is even and not a prime power, 2*a(n) = a(n-1).
If n is odd and not a prime power, 2*n*a(n) = (n+1)*a(n-1). (End)

Links

Crossrefs

Cf. A068550, A180000.

Programs

  • Maple
    a := n -> lcm(seq(k,k=1..n))/binomial(n,iquo(n,2)):
seq(a(n), n=0..49); # Peter Luschny, Oct 23 2015
  • Mathematica
    Join[{1}, Table[LCM @@ Range[n]/Binomial[n, Floor[n/2]], {n, 1, 50}]] (* or *) Table[Product[Cyclotomic[k, 1], {k, 2, n}]/Binomial[n, Floor[n/2]], {n, 0, 50}] (* G. C. Greubel, Apr 17 2017 *)
  • PARI
    A263673(n) = lcm(vector(n,i,i)) / binomial(n,n\2);

Formula

a(n) = A003418(n) / A001405(n).
a(n) = A048619(n-1) * A110654(n).
a(2*n) = A068550(n) = A099996(n) / A000984(n).
a(n) = A180000(n)*A152271(n). - Peter Luschny, Oct 23 2015
a(n) = (e/2)^(n + o(1)). - Charles R Greathouse IV, Oct 23 2015
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.