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A120111 Bi-diagonal inverse matrix of A120108.

%I A120111 #19 Mar 31 2024 17:27:32
%S A120111 1,-2,1,0,-3,1,0,0,-2,1,0,0,0,-5,1,0,0,0,0,-1,1,0,0,0,0,0,-7,1,0,0,0,
%T A120111 0,0,0,-2,1,0,0,0,0,0,0,0,-3,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,
%U A120111 -11,1
%N A120111 Bi-diagonal inverse matrix of A120108.
%C A120111 Subdiagonal is -lcm(1,...,n+2)/lcm(1,...,n+1) or -A014963(n+1).
%C A120111 Row sums are A120112.
%H A120111 G. C. Greubel, <a href="/A120111/b120111.txt">Rows n = 0..100 of the triangle, flattened</a>
%e A120111 Triangle begins
%e A120111    1;
%e A120111   -2,  1;
%e A120111    0, -3,  1;
%e A120111    0,  0, -2,  1;
%e A120111    0,  0,  0, -5,  1;
%e A120111    0,  0,  0,  0, -1,  1;
%e A120111    0,  0,  0,  0,  0, -7,  1;
%e A120111    0,  0,  0,  0,  0,  0, -2,  1;
%e A120111    0,  0,  0,  0,  0,  0,  0, -3,  1;
%e A120111    0,  0,  0,  0,  0,  0,  0,  0, -1,   1;
%e A120111    0,  0,  0,  0,  0,  0,  0,  0,  0, -11, 1;
%t A120111 T[n_, k_] := Switch[k, n, 1, n-1, -Exp[MangoldtLambda[n+1]], _, 0];
%t A120111 Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _Jean-François Alcover_, Mar 01 2021 *)
%t A120111 (* Second program *)
%t A120111 A014963[n_]:= LCM@@Range[n]/(LCM@@Range[n-1]);
%t A120111 A120111[n_, k_]:= If[k==n, 1, If[k==n-1, -A014963[n+1], 0]];
%t A120111 Table[A120111[n,k], {n,0,20}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 05 2023 *)
%o A120111 (Magma)
%o A120111 A014963:= func< n | Lcm([1..n])/Lcm([1..n-1]) >;
%o A120111 A120111:= func< n,k | k eq n select 1 else k eq n-1 select -A014963(n+1) else 0 >;
%o A120111 [A120111(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, May 05 2023
%o A120111 (SageMath)
%o A120111 def A014963(n): return lcm(range(1,n+1))/lcm(range(1,n))
%o A120111 def A120111(n,k):
%o A120111     if (k<n-1): return 0
%o A120111     elif (k==n-1): return -A014963(n+1)
%o A120111     else: return 1
%o A120111 flatten([[A120111(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, May 05 2023
%Y A120111 Cf. A014963, A120108, A120112.
%K A120111 easy,sign,tabl
%O A120111 0,2
%A A120111 _Paul Barry_, Jun 09 2006