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

%I A120101 #19 May 04 2023 08:57:26
%S A120101 1,6,1,30,5,1,420,70,14,3,1260,210,42,9,2,13860,2310,462,99,22,5,
%T A120101 180180,30030,6006,1287,286,65,15,360360,60060,12012,2574,572,130,30,
%U A120101 7,6126120,1021020,204204,43758,9724,2210,510,119,28,116396280,19399380,3879876,831402,184756,41990,9690,2261,532,126
%N A120101 Triangle T(n,k) = lcm(1,...,2*n+2)/((k+1)*binomial(2*k+2,k+1)).
%C A120101 The rows give the coefficients of polynomials arising in the integration of x^(2m)/sqrt(4-x^2), m >= 0.
%H A120101 Muniru A Asiru, <a href="/A120101/b120101.txt">Table of n, a(n) for n = 0..5050</a>
%F A120101 Number triangle T(n,k) = [k<=n] * lcm(1,...,2n+2)/((k+1)*binomial(2k+2, k+1)).
%e A120101 Triangle begins:
%e A120101        1;
%e A120101        6,     1;
%e A120101       30,     5,     1;
%e A120101      420,    70,    14,    3;
%e A120101     1260,   210,    42,    9,   2;
%e A120101    13860,  2310,   462,   99,  22,   5;
%e A120101   180180, 30030,  6006, 1287, 286,  65, 15;
%e A120101   360360, 60060, 12012, 2574, 572, 130, 30, 7;
%p A120101 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
%t A120101 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 *)
%o A120101 (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
%o A120101 (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
%o A120101 (SageMath)
%o A120101 def A120101(n,k):
%o A120101     return lcm(range(1,2*n+3))/((k+1)*(k+2)*catalan_number(k+1))
%o A120101 flatten([[A120101(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, May 03 2023
%Y A120101 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.
%Y A120101 Cf. A000984, A003418.
%K A120101 easy,nonn,tabl
%O A120101 0,2
%A A120101 _Paul Barry_, Jun 09 2006