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%I #9 Sep 08 2022 08:45:24
%S 1,-1,1,-1,0,1,0,0,-1,1,0,-1,0,0,1,0,0,0,0,-1,1,0,0,-1,0,0,0,1,0,0,0,
%T 0,0,0,-1,1,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,
%U 0,0,1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1
%N Expansion of (1-x+x*y)/(1-x^2*y^2) - x^2/(1-x^2*y).
%C Row sums are A000007. Diagonal sums are A115953. Inverse is A115954.
%H G. C. Greubel, <a href="/A115952/b115952.txt">Rows n = 0..100 of triangle, flattened</a>
%F Number triangle T(n,k)=if(n=k,1,0) OR if(n=2k+2,-1,0) OR if(n=k+1,-(1+(-1)^k)/2,0).
%e Triangle begins
%e 1,
%e -1, 1,
%e -1, 0, 1,
%e 0, 0, -1, 1,
%e 0, -1, 0, 0, 1,
%e 0, 0, 0, 0, -1, 1,
%e 0, 0, -1, 0, 0, 0, 1,
%e 0, 0, 0, 0, 0, 0, -1, 1,
%e 0, 0, 0, -1, 0, 0, 0, 0, 1,
%e 0, 0, 0, 0, 0, 0, 0, 0, -1, 1,
%e 0, 0, 0, 0, -1, 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, -1, 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, -1, 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
%t T[n_, k_]:= If[n==k, 1, If[n==k+1, -(1+(-1)^k)/2, If[n==2*k+2, -1, 0]]];
%t Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* _G. C. Greubel_, May 06 2019 *)
%o (PARI) {T(n,k) = if(n==k, 1, if(n==k+1, -(1+(-1)^k)/2, if(n==2*k+2, -1, 0)))}; \\ _G. C. Greubel_, May 06 2019
%o (Magma) [[n eq k select 1 else n eq k+1 select -(1+(-1)^k)/2 else n eq 2*(k+1) select -1 else 0: k in [0..n]]: n in [0..15]]; // _G. C. Greubel_, May 06 2019
%o (Sage)
%o def T(n, k):
%o if (n==k): return 1
%o elif (n==k+1): return -(1+(-1)^k)/2
%o elif (n==2*(k+1)): return -1
%o else: return 0
%o [[T(n, k) for k in (0..n)] for n in (0..15)] # _G. C. Greubel_, May 06 2019
%Y Cf. A115524.
%K easy,sign,tabl
%O 0,1
%A _Paul Barry_, Feb 02 2006
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