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%I #10 Nov 24 2021 11:45:16
%S 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,
%T 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,
%U -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
%N A divide-and-conquer triangle related to A007583.
%C Product of (1-x, x), which is A167374, and number triangle A115715.
%H G. C. Greubel, <a href="/A115717/b115717.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Sum_{k=0..n} T(n, k) = A115716(n).
%F T(n ,k) = Sum_{j=k..n} A167374(n, j)*A115715(j, k). - _R. J. Mathar_, Sep 07 2016
%e Triangle begins
%e 1;
%e 0, 1;
%e 3, -1, 1;
%e 0, 0, 0, 1;
%e 0, 4, -1, -1, 1;
%e 0, 0, 0, 0, 0, 1;
%e 12, -4, 4, 0, -1, -1, 1;
%e 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, 4, 0, 0, -1, -1, 1;
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 16, -4, -4, 4, 0, 0, 0, -1, -1, 1;
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, -1, -1, 1;
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 48, -16, 16, 0, -4, -4, 4, 0, 0, 0, 0, 0, -1, -1, 1;
%p A115717 := proc(n,k)
%p add( A167374(n,j)*A115715(j,k),j=k..n) ;
%p end proc: # _R. J. Mathar_, Sep 07 2016
%t A167374[n_, k_]:= If[k>n-2, (-1)^(n-k), 0];
%t 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 *)
%t f[n_, k_]:= f[n, k]= If[k==n, 1, -Sum[f[n,j]*g[j,k], {j,k+1,n}]]; (* f=A115715 *)
%t A115717[n_, k_]:= A115717[n, k]= Sum[A167374[n,j]*f[j,k], {j,k,n}];
%t Table[A115717[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 23 2021 *)
%o (Sage)
%o @cached_function
%o def A115717(n,k):
%o def A167374(n, k):
%o if (k>n-2): return (-1)^(n-k)
%o else: return 0
%o def A115713(n,k):
%o if (k==n): return 1
%o elif (k==n-1): return -(n%2)
%o elif (n==2*k+2): return -4
%o else: return 0
%o def A115715(n,k):
%o if (k==0): return 4^(floor(log(n+2, 2)) -1)
%o elif (k==n): return 1
%o elif (k==n-1): return (n%2)
%o else: return (-1)*sum( A115715(n,j+k+1)*A115713(j+k+1,k) for j in (0..n-k-1) )
%o return sum( A167374(n, j+k)*A115715(j+k, k) for j in (0..n-k) )
%o flatten([[A115717(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Nov 23 2021
%Y Cf. A007583, A115715, A115716 (row sums), A167374.
%K easy,sign,tabl
%O 0,4
%A _Paul Barry_, Jan 29 2006
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