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 A168646 #17 Apr 06 2025 14:59:53
%S A168646 1,1,1,1,12,1,1,15,15,1,1,16,24,16,1,1,15,30,30,15,1,1,12,30,40,30,12,
%T A168646 1,1,14,63,315,315,63,14,1,1,16,84,224,700,224,84,16,1,1,18,108,336,
%U A168646 630,630,336,108,18,1,1,20,135,480,1050,1512,1050,480,135,20,1,1,22,165,660,1650,2772,2772,1650,660,165,22,1
%N A168646 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,0) = 1, p(x,n) = (8 - n)*(1+x)^n - (7 - n)*(1 + x^n) for 1 <= n <= 6, and p(x,n) = 6*(1+x)^n - Sum_{i=0..4} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n-j))) for n >= 7.
%H A168646 G. C. Greubel, <a href="/A168646/b168646.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A168646 T(n, n-k) = T(n, k). - _G. C. Greubel_, Apr 05 2025
%e A168646 Triangle begins:
%e A168646 1;
%e A168646 1, 1;
%e A168646 1, 12, 1;
%e A168646 1, 15, 15, 1;
%e A168646 1, 16, 24, 16, 1;
%e A168646 1, 15, 30, 30, 15, 1;
%e A168646 1, 12, 30, 40, 30, 12, 1;
%e A168646 1, 14, 63, 315, 315, 63, 14, 1;
%e A168646 1, 16, 84, 224, 700, 224, 84, 16, 1;
%e A168646 1, 18, 108, 336, 630, 630, 336, 108, 18, 1;
%e A168646 1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1;
%e A168646 ...
%t A168646 (* First program *)
%t A168646 p[n_, x_]:= With[{B=Binomial}, If[n==0, 1, If[1<=n<=6, 1 + (8-n)*Sum[B[n,j]*x^j, {j, n -1}] +x^n, Sum[(j+1)*B[n,j]*x^j, {j,0,4}] +6*Sum[B[n,j]*x^j, {j,5,n-5}] + Sum[(n-j+ 1)*B[n,j]*x^j, {j,n-4,n}]]]];
%t A168646 Flatten[Table[CoefficientList[p[n,x], x], {n, 0, 12}]]
%t A168646 (* Second program *)
%t A168646 f[n_, k_]:= If[k==0||k==n,1,If[1<=n<= 6 && 1<=k<=n-1, 8-n, (k+1)*Boole[k<=4] + 6*Boole[5<=k<=n-5] +(n-k+1)*Boole[n-4<=k<=n]]];
%t A168646 A168646[n_, k_]:= Binomial[n,k]*f[n,k];
%t A168646 Table[A168646[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 05 2025 *)
%o A168646 (Maxima) T(n,k) := if k = 0 or k = n then 1 else (if n <= 6 then (8 - n)*binomial(n, k) else ratcoef(6*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 4), x, k))$
%o A168646 create_list(T(n, k), n, 0, 12, k, 0, n); /* _Franck Maminirina Ramaharo_, Jan 02 2019 */
%o A168646 (SageMath)
%o A168646 def f(n,k):
%o A168646 if k==0 or k==n: return 1
%o A168646 elif 0<n<7 and 0<k<n: return 8-n
%o A168646 else: return (k+1)*int(k<5) + 6*int(4<k<n-4) + (n-k+1)*int(k>n-5)
%o A168646 def A168646(n,k): return binomial(n,k)*f(n,k)
%o A168646 print(flatten([[A168646(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Apr 05 2025
%Y A168646 Cf. A132046, A168641, A168643, A168644.
%K A168646 nonn,tabl,easy,less
%O A168646 0,5
%A A168646 _Roger L. Bagula_ and _Gary W. Adamson_, Dec 01 2009
%E A168646 Edited by _Franck Maminirina Ramaharo_, Jan 02 2019
%E A168646 Data values T(7,3), T(7,4), T(8,4) corrected by _G. C. Greubel_, Apr 05 2025