A120105 Number triangle T(n,k) = lcm(1,..,2*n+2)/lcm(1,..,2*k+2).

1, 6, 1, 30, 5, 1, 420, 70, 14, 1, 1260, 210, 42, 3, 1, 13860, 2310, 462, 33, 11, 1, 180180, 30030, 6006, 429, 143, 13, 1, 360360, 60060, 12012, 858, 286, 26, 2, 1, 6126120, 1021020, 204204, 14586, 4862, 442, 34, 17, 1, 116396280, 19399380, 3879876, 277134, 92378, 8398, 646, 323, 19, 1

Offset: 0

Author

Paul Barry, Jun 09 2006

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins:
       1;
       6,     1;
      30,     5,    1;
     420,    70,   14,   1;
    1260,   210,   42,   3,   1;
   13860,  2310,  462,  33,  11,   1;
  180180, 30030, 6006, 429, 143,  13,  1;
		

Links

• Muniru A Asiru, Rows n=0..100 of triangle, flattened

Crossrefs

First column is A119634. Second column is A051538. Inverse is A120111.

Cf. A120106, A120107.

Programs

GAP

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

Magma

[Lcm([1..2*n+2])/Lcm([1..2*k+2]): k in [0..n], n in [0..12]]; // G. C. Greubel, May 04 2023

Maple

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

Mathematica

T[n_, k_]:= LCM@@Range[2*n+2]/(LCM@@Range[2*k+2]);
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 04 2023 *)

SageMath

def f(n): return lcm(range(1,2*n+3))
def A120105(n,k):
    return f(n)//f(k)
flatten([[A120105(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, May 04 2023

Formula

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

From G. C. Greubel, May 04 2023: (Start)

Sum_{k=0..n} T(n, k) = A120106(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A120107(n). (End)