A124039 Triangle read by rows: T(n, k) = (-1)^floor((n+k+2)/2)*(2 - (-1)^(n+k))*A046854(n-1, k-1) with T(1, 1) = 3.

3, 3, -1, -1, -3, 1, -3, 2, 3, -1, 1, 6, -3, -3, 1, 3, -3, -9, 4, 3, -1, -1, -9, 6, 12, -5, -3, 1, -3, 4, 18, -10, -15, 6, 3, -1, 1, 12, -10, -30, 15, 18, -7, -3, 1, 3, -5, -30, 20, 45, -21, -21, 8, 3, -1, -1, -15, 15, 60, -35, -63, 28, 24, -9, -3, 1

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Nov 03 2006

Keywords

• tabl
• sign

Examples

			Triangle begins as:
   3;
   3,  -1;
  -1,  -3,   1;
  -3,   2,   3,  -1;
   1,   6,  -3,  -3,   1;
   3,  -3,  -9,   4,   3,  -1;
  -1,  -9,   6,  12,  -5,  -3,   1;
  -3,   4,  18, -10, -15,   6,   3, -1;
   1,  12, -10, -30,  15,  18,  -7, -3,  1;
   3,  -5, -30,  20,  45, -21, -21,  8,  3, -1;
  -1, -15,  15,  60, -35, -63,  28, 24, -9, -3,  1;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A046854, A066170.

Columns include: (-1)^n*A112030(n-1) (k=1), (-1)^floor((n+1)/2)*A064455(n) (k=2).

Programs

Magma

A124039:= func< n,k | (-1)^Floor((n+k+2)/2)*(2-(-1)^(n+k))*Binomial(Floor((n+k-2)/2), k-1) + 2*0^(n-1) >;
[A124039(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jan 30 2025

Mathematica

(* First program *)
f[n_, m_, d_]:= If[n==m && n>1 && m>1, 0, If[n==m-1 || n==m+1, -1, If[n==m== 1, 3, 0]]];
M[d_]:= Table[T[n,m,d], {n,d}, {m,d}];
A124039[n_]:= Join[{M[1]}, CoefficientList[Det[M[n] - x*IdentityMatrix[n]], x]];
Table[A124039[n], {n,12}]//Flatten
(* Second program *)
A124039[n_, k_]:= (-1)^Floor[(n+k+2)/2]*(2-(-1)^(n-k))*Binomial[Floor[(n+k- 2)/2], k-1] +2*Boole[n==1];
Table[T[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jan 30 2025 *)

SageMath

@CachedFunction
def t(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 3*t(n-1,k) if n==1 else 0
    return t(n-1,k-1) - t(n-2,k) - h
def A124039(n,k): return t(n,k) + 2*0^n
print([[A124039(n,k) for k in range(n+1)] for n in range(13)]) # Peter Luschny, Nov 20 2012

SageMath

def A124039(n,k): return (-1)^((n+k+2)//2)*(2-(-1)^(n+k))*binomial((n+k-2)//2, k-1) + 2*0^(n-1)
print(flatten([[A124039(n,k) for k in range(1,n+1)] for n in range(1,13)])) # G. C. Greubel, Jan 30 2025

Formula

T(n, k) = (-1)^floor((n+k+2)/2)*(2 - (-1)^(n+k))*A046854(n-1,k-1) + 2*[n=1]. - G. C. Greubel, Jan 30 2025

Extensions

Edited by G. C. Greubel, Jan 30 2025