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 A168644 #19 Apr 07 2025 10:07:08
%S A168644 1,1,1,1,10,1,1,12,12,1,1,12,18,12,1,1,10,20,20,10,1,1,12,45,65,45,12,
%T A168644 1,1,14,63,140,154,63,14,1,1,16,84,224,350,252,84,16,1,1,18,108,336,
%U A168644 630,630,384,108,18,1,1,20,135,480,1050,1260,1050,555,135,20,1
%N A168644 Triangle read by rows: T(n, k) = [x^k] p(x,n), where p(x,0) = 1, p(x,n) = (7 - n)*(1+x)^n - (6-n)*(1 + x^n) for 1 <= n <= 5, and p(x,n) = 5*(1 + x)^n - Sum_{i=0..3} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n-j))) + (1/6)*n*(n - 1)*(n - 5)*x^(n-3) for n >= 6.
%H A168644 G. C. Greubel, <a href="/A168644/b168644.txt">Rows n = 0..50 of the triangle, flattened</a>
%e A168644 Triangle begins:
%e A168644 1;
%e A168644 1, 1;
%e A168644 1, 10, 1;
%e A168644 1, 12, 12, 1;
%e A168644 1, 12, 18, 12, 1;
%e A168644 1, 10, 20, 20, 10, 1;
%e A168644 1, 12, 45, 65, 45, 12, 1;
%e A168644 1, 14, 63, 140, 154, 63, 14, 1;
%e A168644 1, 16, 84, 224, 350, 252, 84, 16, 1;
%e A168644 1, 18, 108, 336, 630, 630, 384, 108, 18, 1;
%e A168644 1, 20, 135, 480, 1050, 1260, 1050, 555, 135, 20, 1;
%e A168644 ...
%t A168644 (* First program *)
%t A168644 p[x_, n_]:= If[n<2, (1+x)^n, 5*(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] - If[n>4, x^n +n*x^(n-1) +Binomial[n,2]*(x^2 +x^(n-3) +x^(n-2)) + Binomial[n,3]*x^3 +n*x +1, 1+x^n]];
%t A168644 Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]
%t A168644 (* Second program *)
%t A168644 f[n_, k_]:= With[{B=Boole}, If[k==0 || k==n, 1, If[1<=n<=5, (7-n) - (6-n)*(B[k==0] + B[k==n]), If[n==6, (k+1)*B[k<4] + (n-k+1)*B[k>3] - B[k==3], (k + 1)*B[k<4] + 5*B[3<k<n-3] + (n-k+1)*B[k>n-4]]]]];
%t A168644 A168644[n_, k_]:= Binomial[n,k]*f[n,k] + If[n>5, n*(n-1)*(n-5)*Boole[k==n-3]/6, 0];
%t A168644 Table[A168644[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 06 2025 *)
%o A168644 (Maxima) T(n, k) := if k = 0 or k = n then 1 else (if n <= 5 then (7 - n)*binomial(n, k) else ratcoef(5*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 3) + (1/6)*n*(n - 1)*(n - 5)*x^(n - 3), x, k))$
%o A168644 create_list(T(n, k), n, 0, 12, k, 0, n); /* _Franck Maminirina Ramaharo_, Jan 02 2019 */
%o A168644 (SageMath)
%o A168644 def f(n, k):
%o A168644 if k==0 or k==n: return 1
%o A168644 elif 0<n<6: return 7-n - (6-n)*(int(k==0) + int(k==n))
%o A168644 elif n==6: return (k+1)*int(k<4) + (n-k+1)*int(k>3) - int(k==3)
%o A168644 else: return (k+1)*int(k<4) + 5*int(3<k<n-3) + (n-k+1)*int(k>n-4)
%o A168644 def A168644(n, k):
%o A168644 if n<6: return binomial(n, k)*f(n, k)
%o A168644 else: return binomial(n,k)*f(n,k) + n*(n-1)*(n-5)*int(k==n-3)//6
%o A168644 print(flatten([[A168644(n, k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Apr 06 2025
%Y A168644 Cf. A132046, A168641, A168643, A168646.
%K A168644 nonn,easy,tabl,less
%O A168644 0,5
%A A168644 _Roger L. Bagula_ and _Gary W. Adamson_, Dec 01 2009
%E A168644 Edited by _Franck Maminirina Ramaharo_, Jan 02 2019