A120111 Bi-diagonal inverse matrix of A120108.

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, 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, -11, 1

Offset: 0

Author

Paul Barry, Jun 09 2006

Keywords

• easy
• sign
• tabl

Comments

Subdiagonal is -lcm(1,...,n+2)/lcm(1,...,n+1) or -A014963(n+1).

Row sums are A120112.

Examples

			Triangle begins
   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,  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, -11, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Crossrefs

Cf. A014963, A120108, A120112.

Programs

Magma

A014963:= func< n | Lcm([1..n])/Lcm([1..n-1]) >;
A120111:= func< n,k | k eq n select 1 else k eq n-1 select -A014963(n+1) else 0 >;
[A120111(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, May 05 2023

Mathematica

T[n_, k_] := Switch[k, n, 1, n-1, -Exp[MangoldtLambda[n+1]], _, 0];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* Jean-François Alcover, Mar 01 2021 *)
(* Second program *)
A014963[n_]:= LCM@@Range[n]/(LCM@@Range[n-1]);
A120111[n_, k_]:= If[k==n, 1, If[k==n-1, -A014963[n+1], 0]];
Table[A120111[n,k], {n,0,20}, {k,0,n}]//Flatten (* G. C. Greubel, May 05 2023 *)

SageMath

def A014963(n): return lcm(range(1,n+1))/lcm(range(1,n))
def A120111(n,k):
    if (kA014963(n+1)
    else: return 1
flatten([[A120111(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, May 05 2023