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%I #6 Nov 26 2021 06:08:09
%S 1,1,1,1,0,1,1,1,0,1,1,0,0,0,1,1,1,1,-1,0,1,1,0,0,2,-2,0,1,1,1,0,0,3,
%T -3,0,1,1,0,1,-2,1,4,-4,0,1,1,1,0,3,-6,3,5,-5,0,1,1,0,0,0,6,-12,6,6,
%U -6,0,1,1,1,1,-3,3,9,-20,10,7,-7,0,1,1,0,0,4,-12,12,11,-30,15,8,-8,0,1
%N Triangle T(n,k), read by rows, T(n,k) = (T(n-1,k-1) + T(n-2,k-1)) - (T(n-1,k) + T(n-2,k)), with T(n, 0) = T(n, k) = 1 and T(n, 1) = (n mod 2).
%H G. C. Greubel, <a href="/A174294/b174294.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n,k) = (T(n-1,k-1) + T(n-2,k-1)) - (T(n-1,k) + T(n-2,k)), with T(n, 0) = T(n, k) = 1 and T(n, 1) = (n mod 2).
%e Table begins:
%e n\k|...0...1...2...3...4...5...6...7...8...9..10
%e ---|--------------------------------------------
%e 0..|...1
%e 1..|...1...1
%e 2..|...1...0...1
%e 3..|...1...1...0...1
%e 4..|...1...0...0...0...1
%e 5..|...1...1...1..-1...0...1
%e 6..|...1...0...0...2..-2...0...1
%e 7..|...1...1...0...0...3..-3...0...1
%e 8..|...1...0...1..-2...1...4..-4...0...1
%e 9..|...1...1...0...3..-6...3...5..-5...0...1
%e 10.|...1...0...0...0...6.-12...6...6..-6...0...1
%t T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, If[k==1, Mod[n, 2], T[n-1, k-1] +T[n-2, k-1] -T[n-1, k] -T[n-2, k] ]]];
%t Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 25 2021 *)
%o (Sage)
%o @CachedFunction
%o def T(n,k): # A174294
%o if (k<0 or k>n): return 0
%o elif (k==0 or k==n): return 1
%o elif (k==1): return n%2
%o else: return T(n-1, k-1) + T(n-2, k-1) - T(n-1, k) - T(n-2, k)
%o flatten([[T(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Nov 25 2021
%Y Cf. A112467, A112468, A174294, A174295, A174296, A174297.
%K sign,tabl
%O 0,25
%A _Mats Granvik_, Mar 15 2010
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