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A120113 Bi-diagonal inverse of number triangle A120101.

%I A120113 #10 Mar 31 2024 17:27:44
%S A120113 1,-6,1,0,-5,1,0,0,-14,1,0,0,0,-3,1,0,0,0,0,-11,1,0,0,0,0,0,-13,1,0,0,
%T A120113 0,0,0,0,-2,1,0,0,0,0,0,0,0,-17,1,0,0,0,0,0,0,0,0,-19,1,0,0,0,0,0,0,0,
%U A120113 0,0,-1,1
%N A120113 Bi-diagonal inverse of number triangle A120101.
%C A120113 Subdiagonal is -A120114(n-1).
%H A120113 G. C. Greubel, <a href="/A120113/b120113.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A120113 T(n, k) = 1 if k = n, T(n, k) = -A120114(n-1) if k = n-1, otherwise 0. - _G. C. Greubel_, May 05 2023
%e A120113 Triangle begins
%e A120113    1;
%e A120113   -6,  1;
%e A120113    0, -5,   1;
%e A120113    0,  0, -14,  1;
%e A120113    0,  0,   0, -3,   1;
%e A120113    0,  0,   0,  0, -11,   1;
%e A120113    0,  0,   0,  0,   0, -13,  1;
%e A120113    0,  0,   0,  0,   0,   0, -2,   1;
%e A120113    0,  0,   0,  0,   0,   0,  0, -17,   1;
%e A120113    0,  0,   0,  0,   0,   0,  0,   0, -19,  1;
%e A120113    0,  0,   0,  0,   0,   0,  0,   0,   0, -1,  1;
%t A120113 A120114[n_]:= LCM@@Range[2*n+4]/(LCM@@Range[2*n+2]);
%t A120113 A120113[n_, k_]:= If[k==n, 1, If[k==n-1, -A120114[n-1], 0]];
%t A120113 Table[A120113[n, k], {n,0,16}, {k,0,n}]//Flatten
%o A120113 (Magma)
%o A120113 A120114:= func< n | Lcm([1..2*n+4])/Lcm([1..2*n+2]) >;
%o A120113 A120113:= func< n,k | k eq n select 1 else k eq n-1 select -A120114(n-1) else 0 >;
%o A120113 [A120113(n,k): k in [0..n], n in [0..16]]; // _G. C. Greubel_, May 05 2023
%o A120113 (SageMath)
%o A120113 def A120113(n,k):
%o A120113     if (k<n-1): return 0
%o A120113     elif (k==n-1): return -lcm(range(1,2*n+3))/lcm(range(1,2*n+1))
%o A120113     else: return 1
%o A120113 flatten([[A120113(n,k) for k in range(n+1)] for n in range(17)]) # _G. C. Greubel_, May 05 2023
%Y A120113 Cf. A120101, A120111, A120114.
%K A120113 sign,tabl
%O A120113 0,2
%A A120113 _Paul Barry_, Jun 09 2006