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
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
Programs
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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
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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