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A108086 Triangle, read by rows, where T(0,0) = 1, T(n,k) = (-1)^(n+k)*T(n-1,k) + T(n-1,k-1); a signed version of Pascal's triangle.

%I A108086 #9 Dec 02 2022 07:05:15
%S A108086 1,-1,1,-1,-2,1,1,-3,-3,1,1,4,-6,-4,1,-1,5,10,-10,-5,1,-1,-6,15,20,
%T A108086 -15,-6,1,1,-7,-21,35,35,-21,-7,1,1,8,-28,-56,70,56,-28,-8,1,-1,9,36,
%U A108086 -84,-126,126,84,-36,-9,1,-1,-10,45,120,-210,-252,210,120,-45,-10,1,1,-11,-55,165,330,-462,-462,330,165,-55,-11,1
%N A108086 Triangle, read by rows, where T(0,0) = 1, T(n,k) = (-1)^(n+k)*T(n-1,k) + T(n-1,k-1); a signed version of Pascal's triangle.
%H A108086 G. C. Greubel, <a href="/A108086/b108086.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A108086 T(n,k) = (-1)^(n+k)*T(n-1,k) + T(n-1,k-1), with T(0, 0) = 1.
%F A108086 T(n, k) = (-1)^floor((n-k+1)/2) * A007318(n, k).
%F A108086 From _G. C. Greubel_, Dec 02 2022: (Start)
%F A108086 T(2*n, n) = (-1)^binomial(n+1,2) * A000984(n).
%F A108086 T(2*n, n+1) = (-1)^binomial(n,2) * A001791(n), n >= 1.
%F A108086 T(2*n, n-1) = (-1)^binomial(n+2,2) * A001791(n).
%F A108086 T(2*n+1, n-1) = (-1)^binomial(n-1,2) * A002054(n).
%F A108086 T(2*n+1, n+1) = (-1)^binomial(n+1,2) * A001700(n+1).
%F A108086 Sum_{k=0..n} T(n, k) = (-1)^n * A090132(n).
%F A108086 Sum_{k=0..n} (-1)^k * T(n, k) = A108520(n).
%F A108086 Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n * A260192(n-1).
%F A108086 Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A333378(n+1). (End)
%t A108086 A108086[n_, k_]:= (-1)^(Floor[(n-k+1)/2])*Binomial[n, k];
%t A108086 Table[A108086[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 02 2022 *)
%o A108086 (Magma)
%o A108086 A108086:= func< n,k | (-1)^Floor((n-k+1)/2)*Binomial(n,k) >;
%o A108086 [A108086(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 02 2022
%o A108086 (SageMath)
%o A108086 def A108086(n,k): return (-1)^int((n-k+1)/2)*binomial(n,k)
%o A108086 flatten([[A108086(n,k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Dec 02 2022
%Y A108086 Cf. A000984, A001700, A002054, A007318, A009116.
%Y A108086 Cf. A090132, A108520, A260192, A333378.
%K A108086 sign,tabl
%O A108086 0,5
%A A108086 _Gerald McGarvey_, Jun 05 2005