A115717 A divide-and-conquer triangle related to A007583.
1, 0, 1, 3, -1, 1, 0, 0, 0, 1, 0, 4, -1, -1, 1, 0, 0, 0, 0, 0, 1, 12, -4, 4, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 16, -4, -4, 4, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 48, -16, 16, 0, -4, -4, 4, 0, 0, 0, 0, 0, -1, -1, 1
Offset: 0
Examples
Triangle begins 1; 0, 1; 3, -1, 1; 0, 0, 0, 1; 0, 4, -1, -1, 1; 0, 0, 0, 0, 0, 1; 12, -4, 4, 0, -1, -1, 1; 0, 0, 0, 0, 0, 0, 0, 1; 0, 0, 0, 4, 0, 0, -1, -1, 1; 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; 0, 16, -4, -4, 4, 0, 0, 0, -1, -1, 1; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, -1, -1, 1; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; 48, -16, 16, 0, -4, -4, 4, 0, 0, 0, 0, 0, -1, -1, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Maple
A115717 := proc(n,k) add( A167374(n,j)*A115715(j,k),j=k..n) ; end proc: # R. J. Mathar, Sep 07 2016
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Mathematica
A167374[n_, k_]:= If[k>n-2, (-1)^(n-k), 0]; g[n_, k_]:= g[n, k]= If[k==n, 1, If[k==n-1, -Mod[n, 2], If[n==2*k+2, -4, 0]]]; (* g = A115713 *) f[n_, k_]:= f[n, k]= If[k==n, 1, -Sum[f[n,j]*g[j,k], {j,k+1,n}]]; (* f=A115715 *) A115717[n_, k_]:= A115717[n, k]= Sum[A167374[n,j]*f[j,k], {j,k,n}]; Table[A115717[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 23 2021 *)
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Sage
@cached_function def A115717(n,k): def A167374(n, k): if (k>n-2): return (-1)^(n-k) else: return 0 def A115713(n,k): if (k==n): return 1 elif (k==n-1): return -(n%2) elif (n==2*k+2): return -4 else: return 0 def A115715(n,k): if (k==0): return 4^(floor(log(n+2, 2)) -1) elif (k==n): return 1 elif (k==n-1): return (n%2) else: return (-1)*sum( A115715(n,j+k+1)*A115713(j+k+1,k) for j in (0..n-k-1) ) return sum( A167374(n, j+k)*A115715(j+k, k) for j in (0..n-k) ) flatten([[A115717(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 23 2021
Formula
Sum_{k=0..n} T(n, k) = A115716(n).
Comments