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%I #19 Apr 12 2024 03:26:47
%S 2,2,2,2,-2,2,2,-3,-3,2,2,-4,-6,-4,2,2,-5,-10,-10,-5,2,2,-6,-15,-20,
%T -15,-6,2,2,-7,-21,-35,-35,-21,-7,2,2,-8,-28,-56,-70,-56,-28,-8,2,2,
%U -9,-36,-84,-126,-126,-84,-36,-9,2,2,-10,-45,-120,-210,-252,-210,-120,-45,-10,2
%N Triangle, read by rows, T(n, k) = -binomial(n,k) for 0 < k < n, otherwise T(n, k) = 2.
%C This triangle may also be constructed in the following way. Let f_{n}(t) = d^n/dt^n (t/(1+t) = (-1)^(n+1)*n!*(1+t)^(-n-1). Then the triangle is given as f_{n}(t)/((1+t)*f_{k}(t)*f_{n-k}(t)) when t = 1/2 (this sequence), t = 1/3 (A177228), and t = 1/4 (A177229).
%H G. C. Greubel, <a href="/A177227/b177227.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, 0) = T(n, n) = 2, otherwise T(n, k) = -binomial(n,k).
%F Sum_{k=0..n} T(n, k) = -A131130(n-2) - 3*[n=0], n >= 1 (row sums).
%F From _G. C. Greubel_, Apr 09 2024: (Start)
%F Sum_{k=0..n} (-1)^k*T(n, k) = 3*(1 + (-1)^n) - 4*[n=0].
%F Sum_{k=0..floor(n/2)} T(n-k,k) = (3/2)*(3 + (-1)^n - 2*[n=0])-Fibonacci(n+1).
%F Sum_{k=0..floor(n/2)} (-1)^k*T(n-k,k) = 3*(1 + cos(n*Pi/2) - [n=0]) - (2/sqrt(3))*cos((2*n-1)*Pi/6). (End)
%e Triangle begins as:
%e 2;
%e 2, 2;
%e 2, -2, 2;
%e 2, -3, -3, 2;
%e 2, -4, -6, -4, 2;
%e 2, -5, -10, -10, -5, 2;
%e 2, -6, -15, -20, -15, -6, 2;
%e 2, -7, -21, -35, -35, -21, -7, 2;
%e 2, -8, -28, -56, -70, -56, -28, -8, 2;
%e 2, -9, -36, -84, -126, -126, -84, -36, -9, 2;
%e 2, -10, -45, -120, -210, -252, -210, -120, -45, -10, 2;
%t T[n_, k_]:= If[k==0 || k==n, 2, -Binomial[n,k]];
%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten
%o (Magma)
%o A177227:= func< n,k | k eq 0 or k eq n select 2 else -Binomial(n,k) >;
%o [A177227(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 09 2024
%o (SageMath)
%o def A177227(n,k): return 2 if (k==0 or k==n) else -binomial(n,k)
%o flatten([[A177227(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Apr 09 2024
%Y Cf. A007318, A131130 (related to row sums), A177228, A177229.
%K sign,tabl,less,easy,changed
%O 0,1
%A _Roger L. Bagula_, May 05 2010
%E Edited by _G. C. Greubel_, Apr 09 2024
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