This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A177228 #22 Feb 08 2025 23:35:47
%S A177228 3,3,3,3,-2,3,3,-3,-3,3,3,-4,-6,-4,3,3,-5,-10,-10,-5,3,3,-6,-15,-20,
%T A177228 -15,-6,3,3,-7,-21,-35,-35,-21,-7,3,3,-8,-28,-56,-70,-56,-28,-8,3,3,
%U A177228 -9,-36,-84,-126,-126,-84,-36,-9,3,3,-10,-45,-120,-210,-252,-210,-120,-45,-10
%N A177228 Triangle read by rows: T(n, k) = -binomial(n,k) for 1 <= k <= n-1, otherwise T(n, k) = 3.
%C A177228 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 (this sequence), and t = 1/4 (A177229).
%C A177228 This is the Pascal triangle A007318, with all entries sign-flipped, and 3's inserted at the beginning and end of each row. - _R. J. Mathar_, Mar 27 2024
%H A177228 G. C. Greubel, <a href="/A177228/b177228.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A177228 T(n, k) = -binomial(n,k) for 1 <= k <= n-1, otherwise T(n, k) = 3.
%F A177228 Sum_{k=0..n} T(n, k) = 8 - 2^n, for n >= 1.
%F A177228 From _G. C. Greubel_, Apr 09 2024: (Start)
%F A177228 Sum_{k=0..n} (-1)^k*T(n, k) = 4*(1 + (-1)^n) - 5*[n=0].
%F A177228 Sum_{k=0..floor(n/2)} T(n-k,k) = 2*(3+(-1)^n-2*[n=0])-Fibonacci(n+1).
%F A177228 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k,k) = 4*(1 + cos(n*Pi/2) - [n=0]) - (2/sqrt(3))*cos((2*n-1)*Pi/6). (End)
%e A177228 Triangle begins:
%e A177228 3;
%e A177228 3, 3;
%e A177228 3, -2, 3;
%e A177228 3, -3, -3, 3;
%e A177228 3, -4, -6, -4, 3;
%e A177228 3, -5, -10, -10, -5, 3;
%e A177228 3, -6, -15, -20, -15, -6, 3;
%e A177228 3, -7, -21, -35, -35, -21, -7, 3;
%e A177228 3, -8, -28, -56, -70, -56, -28, -8, 3;
%e A177228 3, -9, -36, -84, -126, -126, -84, -36, -9, 3;
%e A177228 3, -10, -45, -120, -210, -252, -210, -120, -45, -10, 3;
%p A177228 f := proc(n,t)
%p A177228 if n = 0 then
%p A177228 t/(1+t) ;
%p A177228 else
%p A177228 diff( t/(1+t),t$n) ;
%p A177228 factor(%) ;
%p A177228 end if;
%p A177228 end proc:
%p A177228 A177228 := proc(n,m)
%p A177228 f(n,t)/f(m,t)/f(n-m,t) ;
%p A177228 %/(1+t) ;
%p A177228 subs(t=1/3,%) ;
%p A177228 end proc:
%p A177228 seq(seq( A177228(n,m),m=0..n),n=0..12) ; # _R. J. Mathar_, Mar 27 2024
%t A177228 T[n_, k_]:= If[k==0 || k==n, 3, -Binomial[n,k]];
%t A177228 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten
%o A177228 (Magma)
%o A177228 A177228:= func< n,k | k eq 0 or k eq n select 3 else -Binomial(n,k) >;
%o A177228 [A177228(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 09 2024
%o A177228 (SageMath)
%o A177228 def A177228(n,k): return 3 if (k==0 or k==n) else -binomial(n,k)
%o A177228 flatten([[A177228(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Apr 09 2024
%Y A177228 Cf. A007318, A177227, A177229.
%K A177228 sign,tabl,less,easy
%O A177228 0,1
%A A177228 _Roger L. Bagula_, May 05 2010
%E A177228 Edited by _G. C. Greubel_, Apr 09 2024