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

0,2

Comments

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

Links

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

Formula

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

Example

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;

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 *)

Prog

(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
(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

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.

Sequence in context: A118933 A046212 A120105 * A178726 A030524 A327022

Adjacent sequences: A120098 A120099 A120100 * A120102 A120103 A120104

Keyword

easy,nonn,tabl

Author

Paul Barry, Jun 09 2006

Status

approved