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A115633 A divide and conquer-related triangle: see formula for T(n,k), n >= k >= 0.

%I A115633 #11 Nov 24 2021 11:14:39
%S A115633 1,1,-1,-4,0,1,0,0,1,-1,0,-4,0,0,1,0,0,0,0,1,-1,0,0,-4,0,0,0,1,0,0,0,
%T A115633 0,0,0,1,-1,0,0,0,-4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,-4,0,0,0,
%U A115633 0,0,1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,-4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,1
%N A115633 A divide and conquer-related triangle: see formula for T(n,k), n >= k >= 0.
%H A115633 G. C. Greubel, <a href="/A115633/b115633.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A115633 T(n, k) = (-1)^n if n = k; else -4 if n = 2k+2; else (n mod 2) if n = k+1; else 0.
%F A115633 G.f.: (1+x-x*y)/(1-x^2*y^2) - 4*x^2/(1-x^2*y).
%F A115633 (1, -x) + (x, x)/2 + (x, -x)/2 - 4(x^2, x^2) expressed in the notation of stretched Riordan arrays.
%F A115633 Column k has g.f.: (-x)^k + (x*(-x)^k + x^(k+1))/2 - 4*x^(2*k+2).
%F A115633 Sum_{k=0..n} T(n, k) = A115634(n).
%F A115633 Sum_{k=0..floor(n/2)} T(n-k, k) = A115635(n).
%e A115633 Triangle begins
%e A115633    1;
%e A115633    1, -1;
%e A115633   -4,  0,  1;
%e A115633    0,  0,  1, -1;
%e A115633    0, -4,  0,  0,  1;
%e A115633    0,  0,  0,  0,  1, -1;
%e A115633    0,  0, -4,  0,  0,  0,  1;
%e A115633    0,  0,  0,  0,  0,  0,  1, -1;
%e A115633    0,  0,  0, -4,  0,  0,  0,  0,  1;
%e A115633    0,  0,  0,  0,  0,  0,  0,  0,  1, -1;
%e A115633    0,  0,  0,  0, -4,  0,  0,  0,  0,  0,  1;
%e A115633    0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1, -1;
%e A115633    0,  0,  0,  0,  0, -4,  0,  0,  0,  0,  0,  0,  1;
%e A115633    0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1, -1;
%e A115633    0,  0,  0,  0,  0,  0, -4,  0,  0,  0,  0,  0,  0,  0,  1;
%e A115633    0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1, -1;
%e A115633    0,  0,  0,  0,  0,  0,  0, -4,  0,  0,  0,  0,  0,  0,  0,  0, 1;
%t A115633 T[n_, k_]:= If[k==n, (-1)^n, If[k==n-1, (1-(-1)^n)/2, If[n==2*k+2, -4, 0]]];
%t A115633 Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 23 2021 *)
%o A115633 (Sage)
%o A115633 def A115633(n,k):
%o A115633     if (k==n): return (-1)^n
%o A115633     elif (k==n-1): return n%2
%o A115633     elif (n==2*k+2): return -4
%o A115633     else: return 0
%o A115633 flatten([[A115633(n,k) for k in (0..n)] for n in (0..18)]) # _G. C. Greubel_, Nov 23 2021
%o A115633 (PARI) A115633(n,k)=if(n==k, (-1)^n, bittest(n,0), k==n-1, k+1==n\2, -4) \\ _M. F. Hasler_, Nov 24 2021
%Y A115633 Cf. A115634 (row sums), A115635 (diagonal sums), A115636 (inverse).
%K A115633 easy,sign,tabl
%O A115633 0,4
%A A115633 _Paul Barry_, Jan 27 2006