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 A173742 #12 Feb 13 2021 05:06:25
%S A173742 1,1,1,1,8,1,1,9,9,1,1,10,12,10,1,1,11,16,16,11,1,1,12,21,26,21,12,1,
%T A173742 1,13,27,41,41,27,13,1,1,14,34,62,76,62,34,14,1,1,15,42,90,132,132,90,
%U A173742 42,15,1,1,16,51,126,216,258,216,126,51,16,1,1,17,61,171,336,468,468,336,171,61,17,1
%N A173742 Triangle T(n,k) = binomial(n,k) + 6 with T(n,0) = T(n,n) = 1 for n >= 0, read by rows.
%C A173742 For n >= 1, row n sums to A131520(n) + A008586(n).
%H A173742 G. C. Greubel, <a href="/A173742/b173742.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A173742 From _Franck Maminirina Ramaharo_, Dec 09 2018: (Start)
%F A173742 T(n,k) = A007318(n,k) + 6*(1 - A103451(n,k)).
%F A173742 T(n,k) = 7*A007318(n,k) - 6*A132044(n,k).
%F A173742 n-th row polynomial is 3*(1 - (-1)^(2^n)) + (1 + x)^n + 6*(x - x^n)/(1 - x).
%F A173742 G.f.: (1 - (1 + x)*y + 7*x*y^2 - 6*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
%F A173742 E.g.f.: (6 - 6*x + 6*x*exp(y) - 6*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
%F A173742 Sum_{k=0..n} T(n, k) = 2^n + 6*n - 6 + 6*[n=0]. - _G. C. Greubel_, Feb 13 2021
%e A173742 Triangle begins:
%e A173742 1;
%e A173742 1, 1;
%e A173742 1, 8, 1;
%e A173742 1, 9, 9, 1;
%e A173742 1, 10, 12, 10, 1;
%e A173742 1, 11, 16, 16, 11, 1;
%e A173742 1, 12, 21, 26, 21, 12, 1;
%e A173742 1, 13, 27, 41, 41, 27, 13, 1;
%e A173742 1, 14, 34, 62, 76, 62, 34, 14, 1;
%e A173742 1, 15, 42, 90, 132, 132, 90, 42, 15, 1;
%e A173742 1, 16, 51, 126, 216, 258, 216, 126, 51, 16, 1;
%e A173742 ...
%t A173742 T[n_, m_] = Binomial[n, m] + 6*If[m*(n - m) > 0, 1, 0];
%t A173742 Flatten[Table[T[n, m], {n, 0, 10}, {m, 0, n}]]
%o A173742 (Maxima) T(n,k) := if k = 0 or k = n then 1 else binomial(n, k) + 6$
%o A173742 create_list(T(n, k), n, 0, 12, k, 0, n); /* _Franck Maminirina Ramaharo_, Dec 09 2018 */
%o A173742 (Sage)
%o A173742 def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 6
%o A173742 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 13 2021
%o A173742 (Magma)
%o A173742 T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) +6 >;
%o A173742 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 13 2021
%Y A173742 Cf. A007318, A103451, A132044, A156050, A173740, A173741.
%K A173742 nonn,tabl,easy
%O A173742 0,5
%A A173742 _Roger L. Bagula_, Feb 23 2010
%E A173742 Edited and name clarified by _Franck Maminirina Ramaharo_, Dec 09 2018