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

%I A120105 #16 May 05 2023 01:43:39
%S A120105 1,6,1,30,5,1,420,70,14,1,1260,210,42,3,1,13860,2310,462,33,11,1,
%T A120105 180180,30030,6006,429,143,13,1,360360,60060,12012,858,286,26,2,1,
%U A120105 6126120,1021020,204204,14586,4862,442,34,17,1,116396280,19399380,3879876,277134,92378,8398,646,323,19,1
%N A120105 Number triangle T(n,k) = lcm(1,..,2*n+2)/lcm(1,..,2*k+2).
%H A120105 Muniru A Asiru, <a href="/A120105/b120105.txt">Rows n=0..100 of triangle, flattened</a>
%F A120105 Number triangle T(n,k) = [k<=n] + lcm(1,..,2n+2)/lcm(1,..,2k+2).
%F A120105 From _G. C. Greubel_, May 04 2023: (Start)
%F A120105 Sum_{k=0..n} T(n, k) = A120106(n).
%F A120105 Sum_{k=0..floor(n/2)} T(n-k, k) = A120107(n). (End)
%e A120105 Triangle begins:
%e A120105        1;
%e A120105        6,     1;
%e A120105       30,     5,    1;
%e A120105      420,    70,   14,   1;
%e A120105     1260,   210,   42,   3,   1;
%e A120105    13860,  2310,  462,  33,  11,   1;
%e A120105   180180, 30030, 6006, 429, 143,  13,  1;
%p A120105 T:= (n,k)-> ilcm(seq(q,q=1..2*n+2))/ilcm(seq(r,r=1..2*k+2)):
%p A120105 seq(seq(T(n,k),k=0..n),n=0..9); # _Muniru A Asiru_, Feb 26 2019
%t A120105 T[n_, k_]:= LCM@@Range[2*n+2]/(LCM@@Range[2*k+2]);
%t A120105 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 04 2023 *)
%o A120105 (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
%o A120105 (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
%o A120105 (SageMath)
%o A120105 def f(n): return lcm(range(1,2*n+3))
%o A120105 def A120105(n,k):
%o A120105     return f(n)//f(k)
%o A120105 flatten([[A120105(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, May 04 2023
%Y A120105 First column is A119634. Second column is A051538. Inverse is A120111.
%Y A120105 Cf. A120106, A120107.
%K A120105 easy,nonn,tabl
%O A120105 0,2
%A A120105 _Paul Barry_, Jun 09 2006