cp's OEIS Frontend

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.

%I A124039 #17 Jan 30 2025 08:22:46
%S A124039 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,
%T A124039 1,-3,4,18,-10,-15,6,3,-1,1,12,-10,-30,15,18,-7,-3,1,3,-5,-30,20,45,
%U A124039 -21,-21,8,3,-1,-1,-15,15,60,-35,-63,28,24,-9,-3,1
%N 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.
%H A124039 G. C. Greubel, <a href="/A124039/b124039.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A124039 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
%e A124039 Triangle begins as:
%e A124039    3;
%e A124039    3,  -1;
%e A124039   -1,  -3,   1;
%e A124039   -3,   2,   3,  -1;
%e A124039    1,   6,  -3,  -3,   1;
%e A124039    3,  -3,  -9,   4,   3,  -1;
%e A124039   -1,  -9,   6,  12,  -5,  -3,   1;
%e A124039   -3,   4,  18, -10, -15,   6,   3, -1;
%e A124039    1,  12, -10, -30,  15,  18,  -7, -3,  1;
%e A124039    3,  -5, -30,  20,  45, -21, -21,  8,  3, -1;
%e A124039   -1, -15,  15,  60, -35, -63,  28, 24, -9, -3,  1;
%t A124039 (* First program *)
%t A124039 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]]];
%t A124039 M[d_]:= Table[T[n,m,d], {n,d}, {m,d}];
%t A124039 A124039[n_]:= Join[{M[1]}, CoefficientList[Det[M[n] - x*IdentityMatrix[n]], x]];
%t A124039 Table[A124039[n], {n,12}]//Flatten
%t A124039 (* Second program *)
%t A124039 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];
%t A124039 Table[T[n,k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Jan 30 2025 *)
%o A124039 (SageMath)
%o A124039 @CachedFunction
%o A124039 def t(n,k):
%o A124039     if n< 0: return 0
%o A124039     if n==0: return 1 if k == 0 else 0
%o A124039     h = 3*t(n-1,k) if n==1 else 0
%o A124039     return t(n-1,k-1) - t(n-2,k) - h
%o A124039 def A124039(n,k): return t(n,k) + 2*0^n
%o A124039 print([[A124039(n,k) for k in range(n+1)] for n in range(13)]) # _Peter Luschny_, Nov 20 2012
%o A124039 (SageMath)
%o A124039 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)
%o A124039 print(flatten([[A124039(n,k) for k in range(1,n+1)] for n in range(1,13)])) # _G. C. Greubel_, Jan 30 2025
%o A124039 (Magma)
%o A124039 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) >;
%o A124039 [A124039(n, k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jan 30 2025
%Y A124039 Cf. A046854, A066170.
%Y A124039 Columns include: (-1)^n*A112030(n-1) (k=1), (-1)^floor((n+1)/2)*A064455(n) (k=2).
%K A124039 tabl,sign
%O A124039 1,1
%A A124039 _Roger L. Bagula_ and _Gary W. Adamson_, Nov 03 2006
%E A124039 Edited by _G. C. Greubel_, Jan 30 2025