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A divide and conquer-related triangle: see formula for T(n,k), n >= k >= 0.
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%I #11 Nov 24 2021 11:14:39

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

%H G. C. Greubel, <a href="/A115633/b115633.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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 G.f.: (1+x-x*y)/(1-x^2*y^2) - 4*x^2/(1-x^2*y).

%F (1, -x) + (x, x)/2 + (x, -x)/2 - 4(x^2, x^2) expressed in the notation of stretched Riordan arrays.

%F Column k has g.f.: (-x)^k + (x*(-x)^k + x^(k+1))/2 - 4*x^(2*k+2).

%F Sum_{k=0..n} T(n, k) = A115634(n).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = A115635(n).

%e Triangle begins

%e 1;

%e 1, -1;

%e -4, 0, 1;

%e 0, 0, 1, -1;

%e 0, -4, 0, 0, 1;

%e 0, 0, 0, 0, 1, -1;

%e 0, 0, -4, 0, 0, 0, 1;

%e 0, 0, 0, 0, 0, 0, 1, -1;

%e 0, 0, 0, -4, 0, 0, 0, 0, 1;

%e 0, 0, 0, 0, 0, 0, 0, 0, 1, -1;

%e 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 1;

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1;

%e 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 1;

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1;

%e 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 1;

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1;

%e 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 1;

%t T[n_, k_]:= If[k==n, (-1)^n, If[k==n-1, (1-(-1)^n)/2, If[n==2*k+2, -4, 0]]];

%t Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 23 2021 *)

%o (Sage)

%o def A115633(n,k):

%o if (k==n): return (-1)^n

%o elif (k==n-1): return n%2

%o elif (n==2*k+2): return -4

%o else: return 0

%o flatten([[A115633(n,k) for k in (0..n)] for n in (0..18)]) # _G. C. Greubel_, Nov 23 2021

%o (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 Cf. A115634 (row sums), A115635 (diagonal sums), A115636 (inverse).

%K easy,sign,tabl

%O 0,4

%A _Paul Barry_, Jan 27 2006