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A172090 Triangle T(n, k) = f(n-k) + f(k) - f(n), where f(n) = -3*n with f(0) = 1, f(1) = -2, read by rows.

%I A172090 #20 Apr 25 2024 09:17:24
%S A172090 1,1,1,1,2,1,1,1,1,1,1,1,0,1,1,1,1,0,0,1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,
%T A172090 1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,
%U A172090 0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1
%N A172090 Triangle T(n, k) = f(n-k) + f(k) - f(n), where f(n) = -3*n with f(0) = 1, f(1) = -2, read by rows.
%H A172090 G. C. Greubel, <a href="/A172090/b172090.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A172090 T(n, k) = f(n-k) + f(k) - f(n), where f(n) = -3*n with f(0) = 1, f(1) = -2.
%F A172090 From _G. C. Greubel_, Apr 29 2021: (Start)
%F A172090 T(n, k) is defined by T(n, 0) = T(n, 1) = T(n, n-1) = T(n, n) = T(3, k) = 1, T(2, 1) = 2 and 0 otherwise.
%F A172090 Sum_{k=0..n} T(n,k) = A151798(n). (End)
%e A172090 Triangle begins as:
%e A172090   1;
%e A172090   1, 1;
%e A172090   1, 2, 1;
%e A172090   1, 1, 1, 1;
%e A172090   1, 1, 0, 1, 1;
%e A172090   1, 1, 0, 0, 1, 1;
%e A172090   1, 1, 0, 0, 0, 1, 1;
%e A172090   1, 1, 0, 0, 0, 0, 1, 1;
%e A172090   1, 1, 0, 0, 0, 0, 0, 1, 1;
%e A172090   1, 1, 0, 0, 0, 0, 0, 0, 1, 1;
%e A172090   1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1;
%t A172090 (* First program *)
%t A172090 f[n_]:= f[n]= If[n < 2, (-1)^n*(n+1), -3*n];
%t A172090 T[n_, k_]:= f[n-k] +f[k] -f[n];
%t A172090 Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* modified by _G. C. Greubel_, Apr 29 2021 *)
%t A172090 (* Second program *)
%t A172090 T[n_, k_]:= If[n<3, Binomial[n, k], If[n==3 || k<2 || k>n-2, 1, 0]];
%t A172090 Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Apr 29 2021 *)
%o A172090 (Sage)
%o A172090 def f(n): return (-1)^n*(n+1) if (n<2) else -3*n
%o A172090 def T(n,k): return f(n-k) + f(k) - f(n)
%o A172090 flatten([[T(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Apr 29 2021
%Y A172090 Row sums are A151798.
%K A172090 nonn,tabl,easy,less
%O A172090 0,5
%A A172090 _Roger L. Bagula_, Jan 25 2010
%E A172090 Edited by _G. C. Greubel_, Apr 29 2021