A120113 -id:A120113 - cp's OEIS frontend

A120101 Triangle T(n,k) = lcm(1,...,2n+2)/((k+1)binomial(2*k+2,k+1)).

1, 6, 1, 30, 5, 1, 420, 70, 14, 3, 1260, 210, 42, 9, 2, 13860, 2310, 462, 99, 22, 5, 180180, 30030, 6006, 1287, 286, 65, 15, 360360, 60060, 12012, 2574, 572, 130, 30, 7, 6126120, 1021020, 204204, 43758, 9724, 2210, 510, 119, 28, 116396280, 19399380, 3879876, 831402, 184756, 41990, 9690, 2261, 532, 126

Offset: 0

Paul Barry, Jun 09 2006

The rows give the coefficients of polynomials arising in the integration of x^(2m)/sqrt(4-x^2), m >= 0.

			Triangle begins:
       1;
       6,     1;
      30,     5,     1;
     420,    70,    14,    3;
    1260,   210,    42,    9,   2;
   13860,  2310,   462,   99,  22,   5;
  180180, 30030,  6006, 1287, 286,  65, 15;
  360360, 60060, 12012, 2574, 572, 130, 30, 7;

Muniru A Asiru, Table of n, a(n) for n = 0..5050

First column is A119634. Second column is A051538. Main diagonal is A068553. Subdiagonal is A119636. Inverse is A120113. Row sums are A120106. Diagonal sums are A120107.

Cf. A000984, A003418.

Flat(List([0..9],n->List([0..n],k->Lcm(List([1..2*n+2],i->i))/((k+1)*Binomial(2*k+2,k+1))))); # Muniru A Asiru, Feb 26 2019

[Lcm([1..2*n+2])/((k+1)*(k+2)*Catalan(k+1)): k in [0..n], n in [0..12]]; // G. C. Greubel, May 03 2023

T:=(n,k)-> ilcm(seq(q,q=1..2*n+2))/((k+1)*binomial(2*k+2,k+1)): seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Feb 26 2019

Table[LCM@@Range[2*n+2]/((k+1)*Binomial[2*k+2,k+1]), {n,0,12}, {k,0, n}]//Flatten (* G. C. Greubel, May 03 2023 *)

def A120101(n,k):
    return lcm(range(1,2*n+3))/((k+1)*(k+2)*catalan_number(k+1))
flatten([[A120101(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, May 03 2023

Number triangle T(n,k) = [k<=n] * lcm(1,...,2n+2)/((k+1)*binomial(2k+2, k+1)).

A120114 a(n) = lcm(1, ..., 2n+4)/lcm(1, ..., 2n+2).

Original entry on oeis.org

6, 5, 14, 3, 11, 13, 2, 17, 19, 1, 23, 5, 3, 29, 62, 1, 1, 37, 1, 41, 43, 1, 47, 7, 1, 53, 1, 1, 59, 61, 2, 1, 67, 1, 71, 73, 1, 1, 79, 3, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 11, 1, 5, 254, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1

Offset: 0

Paul Barry, Jun 09 2006

The subdiagonal of A120113 is -a(n).
From Robert Israel, Dec 03 2024: (Start)
a(n) is the product of the primes p such that 2*n + 3 or 2*n + 4 is a power of p.
Thus: a(n) = 1 if and only if neither 2*n + 3 nor 2*n + 4 is in A000961.
      if n + 1 = 2^k - 1 is a Mersenne number but not a Mersenne prime, then a(n) = 2;
      if n + 1 = 2^k - 1 is a Mersenne prime, then a(n) = 2 * (2^k - 1);
      otherwise a(n) is odd. (End)
Conjectures from Davide Rotondo, Dec 02 2024: (Start)
Except for 2, if a(n) is even then a(n)/2 is a Mersenne prime.
If a(n)=1 or a(n)=2 then (n*2)+3 is in A061346, or also, or (n+1) is in A083390. (End)

Muniru A Asiru, Table of n, a(n) for n = 0..5000

Cf. A000961, A099996, A120113.

List([0..75],n->Lcm(List([1..2*n+4],i->i))/Lcm(List([1..2*n+2],i->i))); # Muniru A Asiru, Mar 04 2019

A120114:= func< n | Lcm([1..2*n+4])/Lcm([1..2*n+2]) >;
[A120114(n): n in [0..100]]; // G. C. Greubel, May 05 2023

f:= proc(n) local t,x,S;
   t:= 1;
   for x from 2*n+3 to 2*n+4 do
     S:= numtheory:-factorset(x);
     if nops(S) = 1 then t:= t*S[1] fi;
   od;
   t
end proc:
map(f, [$0..100]); # Robert Israel, Dec 03 2024

Table[(LCM@@Range[2n+4])/LCM@@Range[2n+2],{n,0,100}] (* Harvey P. Dale, Dec 15 2017 *)

def A120114(n):
    return lcm(range(1,2*n+5)) // lcm(range(1,2*n+3))
[A120114(n) for n in range(101)] # G. C. Greubel, May 05 2023

a(n) = A099996(n+2)/A099996(n+1). - Michel Marcus, May 06 2023

More terms from Harvey P. Dale, Dec 15 2017

cp's OEIS Frontend

A120101 Triangle T(n,k) = lcm(1,...,2n+2)/((k+1)binomial(2*k+2,k+1)).

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A120114 a(n) = lcm(1, ..., 2n+4)/lcm(1, ..., 2n+2).

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cp's OEIS Frontend

A120101 Triangle T(n,k) = lcm(1,...,2*n+2)/((k+1)*binomial(2*k+2,k+1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

SageMath

Formula

A120114 a(n) = lcm(1, ..., 2n+4)/lcm(1, ..., 2n+2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

SageMath

Formula

Extensions

A120101 Triangle T(n,k) = lcm(1,...,2n+2)/((k+1)binomial(2*k+2,k+1)).