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A115636 A divide-and-conquer number triangle.

%I A115636 #6 Nov 24 2021 10:40:54
%S A115636 1,1,-1,4,0,1,4,0,1,-1,4,-4,0,0,1,4,-4,0,0,1,-1,16,0,4,0,0,0,1,16,0,4,
%T A115636 0,0,0,1,-1,16,0,4,-4,0,0,0,0,1,16,0,4,-4,0,0,0,0,1,-1,16,-16,0,0,4,0,
%U A115636 0,0,0,0,1,16,-16,0,0,4,0,0,0,0,0,1,-1,16,-16,0,0,4,-4,0,0,0,0,0,0,1,16,-16,0,0,4,-4,0,0,0,0,0,0,1,-1
%N A115636 A divide-and-conquer number triangle.
%H A115636 G. C. Greubel, <a href="/A115636/b115636.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A115636 T(n, 0) = A115639(n).
%F A115636 Sum_{k=0..n} T(n, k) = A115637(n).
%F A115636 T(n, k) = (-1)^k*( 1 if k = n otherwise (-1)*Sum_{j=k+1..n} T(n, j)*A115633(j, k) ). - _G. C. Greubel_, Nov 24 2021
%e A115636 Triangle begins
%e A115636    1;
%e A115636    1,  -1;
%e A115636    4,   0,  1;
%e A115636    4,   0,  1, -1;
%e A115636    4,  -4,  0,  0,  1;
%e A115636    4,  -4,  0,  0,  1, -1;
%e A115636   16,   0,  4,  0,  0,  0,  1;
%e A115636   16,   0,  4,  0,  0,  0,  1, -1;
%e A115636   16,   0,  4, -4,  0,  0,  0,  0,  1;
%e A115636   16,   0,  4, -4,  0,  0,  0,  0,  1, -1;
%e A115636   16, -16,  0,  0,  4,  0,  0,  0,  0,  0,  1;
%e A115636   16, -16,  0,  0,  4,  0,  0,  0,  0,  0,  1, -1;
%e A115636   16, -16,  0,  0,  4, -4,  0,  0,  0,  0,  0,  0,  1;
%e A115636   16, -16,  0,  0,  4, -4,  0,  0,  0,  0,  0,  0,  1, -1;
%e A115636   64,   0, 16,  0,  0,  0,  4,  0,  0,  0,  0,  0,  0,  0,  1;
%e A115636   64,   0, 16,  0,  0,  0,  4,  0,  0,  0,  0,  0,  0,  0,  1, -1;
%e A115636   64,   0, 16,  0,  0,  0,  4, -4,  0,  0,  0,  0,  0,  0,  0,  0,  1;
%t A115636 A115633[n_, k_]:= If[k==n, (-1)^n, If[k==n-1, Mod[n,2], If[n==2*k+2, -4, 0]]];
%t A115636 T[n_, k_]:= T[n, k]= (-1)^k*If[k==n, 1, -Sum[T[n, j]*A115633[j, k], {j,k+1,n}] ];
%t A115636 Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 24 2021 *)
%o A115636 (Sage)
%o A115636 @CachedFunction
%o A115636 def A115633(n, k):
%o A115636     if (k==n): return (-1)^n
%o A115636     elif (k==n-1): return n%2
%o A115636     elif (n==2*k+2): return -4
%o A115636     else: return 0
%o A115636 def A115636(n,k):
%o A115636     if (k==0): return 4^(floor(log(n+2, 2)) -1)
%o A115636     elif (k==n): return (-1)^n
%o A115636     elif (k==n-1): return (n%2)
%o A115636     else: return (-1)^(k+1)*sum( A115636(n, j)*A115633(j, k) for j in (k+1..n) )
%o A115636 flatten([[A115636(n, k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Nov 24 2021
%Y A115636 Cf. A115633 (inverse), A115637 (row sums), A115639 (first column).
%K A115636 sign,tabl
%O A115636 0,4
%A A115636 _Paul Barry_, Jan 27 2006