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

Author

Paul Barry, Jun 09 2006

Keywords

• easy
• nonn
• tabl

Comments

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

Examples

			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;
		

Links

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

Crossrefs

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.

Programs

GAP

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

Magma

[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

Maple

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

Mathematica

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 *)

SageMath

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

Formula

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