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%I #16 Apr 12 2024 03:26:16
%S 4,4,4,4,-2,4,4,-3,-3,4,4,-4,-6,-4,4,4,-5,-10,-10,-5,4,4,-6,-15,-20,
%T -15,-6,4,4,-7,-21,-35,-35,-21,-7,4,4,-8,-28,-56,-70,-56,-28,-8,4,4,
%U -9,-36,-84,-126,-126,-84,-36,-9,4,4,-10,-45,-120,-210,-252,-210,-120,-45,-10,4
%N Triangle, read by rows, T(n, k) = -binomial(n,k) for 1 <= k <= n-1, otherwise T(n, k) = 4.
%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 (A177227), t = 1/3 (A177228), and t = 1/4 (this sequence).
%H G. C. Greubel, <a href="/A177229/b177229.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = -binomial(n,k) for 1 <= k <= n-1, otherwise T(n, k) = 4.
%F From _G. C. Greubel_, Apr 09 2024: (Start)
%F Sum_{k=0..n} T(n, k) = 10 - 2^n - 5*[n=0] (row sums).
%F Sum_{k=0..n} (-1)^k*T(n, k) = 5*(1 + (-1)^n) - 6*[n=0].
%F Sum_{k=0..floor(n/2)} T(n-k,k) = (5/2)*(3+(-1)^n-2*[n=0])-Fibonacci(n+1).
%F Sum_{k=0..floor(n/2)} (-1)^k*T(n-k,k) = 5*(1 + cos(n*Pi/2) - [n=0]) - (2/sqrt(3))*cos((2*n-1)*Pi/6). (End)
%e Triangle begins:
%e 4;
%e 4, 4;
%e 4, -2, 4;
%e 4, -3, -3, 4;
%e 4, -4, -6, -4, 4;
%e 4, -5, -10, -10, -5, 4;
%e 4, -6, -15, -20, -15, -6, 4;
%e 4, -7, -21, -35, -35, -21, -7, 4;
%e 4, -8, -28, -56, -70, -56, -28, -8, 4;
%e 4, -9, -36, -84, -126, -126, -84, -36, -9, 4;
%e 4, -10, -45, -120, -210, -252, -210, -120, -45, -10, 4;
%t T[n_, k_]:= If[k==0 || k==n, 4, -Binomial[n,k]];
%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten
%o (Magma)
%o A177229:= func< n,k | k eq 0 or k eq n select 4 else -Binomial(n,k) >;
%o [A177229(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 09 2024
%o (SageMath)
%o def A177229(n,k): return 4 if (k==0 or k==n) else -binomial(n,k)
%o flatten([[A177229(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Apr 09 2024
%Y Cf. A007318, A177227, A177228.
%K sign,tabl,less,easy
%O 0,1
%A _Roger L. Bagula_, May 05 2010
%E Edited by _G. C. Greubel_, Apr 09 2024
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