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 A168641 #18 Apr 05 2025 23:39:24
%S A168641 1,1,1,1,2,1,1,6,6,1,1,8,18,8,1,1,10,30,30,10,1,1,12,45,60,45,12,1,1,
%T A168641 14,63,105,105,63,14,1,1,16,84,168,210,168,84,16,1,1,18,108,252,378,
%U A168641 378,252,108,18,1,1,20,135,360,630,756,630,360,135,20,1
%N A168641 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,n) = 3*(x + 1)^n - 2*(x^n + 1) - n*(x + x^(n - 1)) for n >= 2, p(x,0) = 1, and p(x,1) = x + 1.
%H A168641 G. C. Greubel, <a href="/A168641/b168641.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A168641 From _G. C. Greubel_, Mar 24 2025: (Start)
%F A168641 T(n, k) = 3*binomial(n, k), for n >= 4 and 2 <= k <= n-2, otherwise T(n, 0) = T(n, n) = 1, T(n, 1) = T(n, n-1) = 2*A065475(n-1).
%F A168641 T(n, n-k) = T(n, k).
%F A168641 T(n, 1) = A005843(n) - [n=1] - 2*[n=2].
%F A168641 Columns: T(n, k) = 3*binomial(n,k) - 2*[n=k] - (k+1)*[n=k+1], k >= 2.
%F A168641 Sum_{k=0..n} T(n, k) = 2*A095151(n-1) - 2*[n=0] - 2*[n=1].
%F A168641 Sum_{k=0..n} (-1)^k*T(n, k) = (1+(-1)^n)*(n-2) + 5*[n=0]. (End)
%e A168641 Triangle begins:
%e A168641 1;
%e A168641 1, 1;
%e A168641 1, 2, 1;
%e A168641 1, 6, 6, 1;
%e A168641 1, 8, 18, 8, 1;
%e A168641 1, 10, 30, 30, 10, 1;
%e A168641 1, 12, 45, 60, 45, 12, 1;
%e A168641 1, 14, 63, 105, 105, 63, 14, 1;
%e A168641 1, 16, 84, 168, 210, 168, 84, 16, 1;
%e A168641 1, 18, 108, 252, 378, 378, 252, 108, 18, 1;
%e A168641 1, 20, 135, 360, 630, 756, 630, 360, 135, 20, 1;
%e A168641 ...
%t A168641 p[x_, n_]:= If[n==0, 1, If[n==1, 1+x, 3*(1+x)^n -(1+x^n) -(1+n*x +n*x^(n-1) + x^n)]];
%t A168641 Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]
%t A168641 (* Second program *)
%t A168641 f[n_, k_]:= With[{b=Boole}, If[k<=n/2, b[k==0] +2*b[k==1] +3*b[2<=k<=n/2], f[n, n-k]]];
%t A168641 A168641[n_, k_]:= Binomial[n,k]*If[n<3,1,f[n,k]];
%t A168641 Table[A168641[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 24 2025 *)
%o A168641 (Maxima) T(n,k) := ratcoef(if n <= 2 then (1 + x)^n else 3*(x + 1)^n - (x^n + 1) - (x^n + n*x^(n - 1) + n*x + 1), x, k);
%o A168641 create_list(T(n, k), n, 0, 12, k, 0, n); /* _Franck Maminirina Ramaharo_, Jan 02 2019 */
%o A168641 (Magma)
%o A168641 function f(n,k)
%o A168641 if n le 2 then return 1;
%o A168641 elif k eq 0 or k eq n then return 1;
%o A168641 elif k eq 1 or k eq n-1 then return 2;
%o A168641 else return 3;
%o A168641 end if;
%o A168641 end function;
%o A168641 A168641:= func< n,k | Binomial(n,k)*f(n,k) >;
%o A168641 [A168641(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 24 2025
%o A168641 (SageMath)
%o A168641 def f(n,k):
%o A168641 if (k<=n/2): return int(k==0) + 2*int(k==1) + 3*int(1<k<=n//2)
%o A168641 else: return f(n, n-k)
%o A168641 def A168641(n,k):
%o A168641 if (n<3): return binomial(n,k)
%o A168641 else: return binomial(n,k)*f(n,k)
%o A168641 print(flatten([[A168641(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Mar 24 2025
%Y A168641 Cf. A095151, A132046, A168643, A168644, A168646.
%Y A168641 Columns (essentially): A005843 (k=1), A045943 (k=2), A027480 (k=3), A050534 (k=4), A253942 (k=5), A253943 (k=6), A253944 (k=7).
%K A168641 nonn,easy,less,tabl
%O A168641 0,5
%A A168641 _Roger L. Bagula_ and _Gary W. Adamson_, Dec 01 2009
%E A168641 Edited by _Franck Maminirina Ramaharo_, Jan 02 2019