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 A168643 #21 Jun 10 2025 23:33:25
%S A168643 1,1,1,1,8,1,1,9,9,1,1,8,12,8,1,1,10,30,30,10,1,1,12,45,80,45,12,1,1,
%T A168643 14,63,140,140,63,14,1,1,16,84,224,280,224,84,16,1,1,18,108,336,504,
%U A168643 504,336,108,18,1,1,20,135,480,840,1008,840,480,135,20,1
%N A168643 Triangle read by rows: T(n,k) = [x^k] p(x,n) where p(x,0) = 1, p(x,n) = (6 - n)*(1+x)^n - (5-n)*(1 + x^n) for 1 <= n <= 4, and p (x,n) = 4*(1+x)^n - Sum_{i=0..2} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n-j))) for n >= 5.
%H A168643 G. C. Greubel, <a href="/A168643/b168643.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A168643 From _G. C. Greubel_, Apr 08 2025: (Start)
%F A168643 T(n, k) = [k=0] + (6-n)*binomial(n,k)*[1 <= k <= n-1] + [k=n] if 1 <= n <= 4, T(n, k) = binomial(n,k)*( (k+1)*[k<3] + 4*[2 < k < n-2] + (n-k+1)*[k > n-3] ) if n >= 5, with T(n, 0) = T(n, n) = 1.
%F A168643 T(n, n-k) = T(n, k) (symmetric rows).
%F A168643 Sum_{k=0..n} T(n, k) = 2^(n+2) - n^2 - 3*n - 6 + 13*[n=3] + 10*[n=2] + 4*[n=1] + 3*[n=0]. (End)
%e A168643 Triangle begins:
%e A168643 1;
%e A168643 1, 1;
%e A168643 1, 8, 1;
%e A168643 1, 9, 9, 1;
%e A168643 1, 8, 12, 8, 1;
%e A168643 1, 10, 30, 30, 10, 1;
%e A168643 1, 12, 45, 80, 45, 12, 1;
%e A168643 1, 14, 63, 140, 140, 63, 14, 1;
%e A168643 1, 16, 84, 224, 280, 224, 84, 16, 1;
%e A168643 1, 18, 108, 336, 504, 504, 336, 108, 18, 1;
%e A168643 1, 20, 135, 480, 840, 1008, 840, 480, 135, 20, 1;
%e A168643 ...
%t A168643 (* First program *)
%t A168643 p[x_, n_]:= If[n==0, 1, If[n==1, x+1, 4*(1+x)^n - (1+x^n) - If[n>2, x^n + n*x^(n-1) +n*x+1, 1+x^n] - If[n>3, x^n +n*x^(n-1) + Binomial[n,2]*(x^2 +x^(n-2)) +n*x+1, 1+x^n]]];
%t A168643 Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]
%t A168643 (* Second program *)
%t A168643 f[n_,k_]:= With[{B=Boole}, If[n==0, 1, If[0<n<5, B[k==0] +(6-n)*B[0<k<n] +B[k==n], (k+1)*B[k<3] +4*B[2<k<n-2] +(n-k+1)*B[k>n-3]]]];
%t A168643 A168643[n_,k_]:= Binomial[n,k]*f[n,k];
%t A168643 Table[A168643[n,k], {n,0,13}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 08 2025 *)
%o A168643 (Maxima) T(n,k) := if k = 0 or k = n then 1 else (if n <= 4 then (6 - n)*binomial(n, k) else ratcoef(4*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 2), x, k))$
%o A168643 create_list(T(n, k), n, 0, 12, k, 0, n); /* _Franck Maminirina Ramaharo_, Jan 02 2019 */
%o A168643 (SageMath)
%o A168643 def f(n,k):
%o A168643 if n==0: return 1
%o A168643 elif 0<n<5: return int(k==0) + (6-n)*int(0<k<n) + int(k==n)
%o A168643 else: return (k+1)*int(k<3) + 4*int(2<k<n-2) + (n-k+1)*int(k>n-3)
%o A168643 def A168643(n,k): return binomial(n,k)*f(n,k)
%o A168643 print(flatten([[A168643(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Apr 08 2025
%Y A168643 Cf. A132046, A168641, A168644, A168646.
%K A168643 nonn,easy,tabl,less
%O A168643 0,5
%A A168643 _Roger L. Bagula_ and _Gary W. Adamson_, Dec 01 2009
%E A168643 Edited by _Franck Maminirina Ramaharo_, Jan 02 2019