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%I #14 Feb 14 2021 01:37:04
%S 1,1,1,1,4,1,1,5,5,1,1,6,8,6,1,1,7,12,12,7,1,1,8,17,22,17,8,1,1,9,23,
%T 37,37,23,9,1,1,10,30,58,72,58,30,10,1,1,11,38,86,128,128,86,38,11,1,
%U 1,12,47,122,212,254,212,122,47,12,1,1,13,57,167,332,464,464,332,167,57,13,1
%N Triangle T(n,k) = binomial(n,k) + 2 for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.
%C For n >= 1, row n sums to A131520(n).
%H G. C. Greubel, <a href="/A173740/b173740.txt">Rows n = 0..100 of the triangle, flattened</a>
%F From _Franck Maminirina Ramaharo_, Dec 08 2018:(Start)
%F T(n,k) = A007318(n,k) + 2*(1 - A103451(n,k)).
%F T(n,k) = 3*A007318(n,k) - 2*A132044(n,k).
%F n-th row polynomial is 1 - (-1)^(2^n) + (1 + x)^n + 2*(x - x^n)/(1 - x).
%F G.f.: (1 - (1 + x)*y + 3*x*y^2 - 2*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
%F E.g.f.: (2 - 2*x + 2*x*exp(y) - 2*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
%F Sum_{k=0..n} T(n, k) = 2^n + 2*(n - 1 + [n=0]) = 2*A100314(n). - _G. C. Greubel_, Feb 13 2021
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 5, 5, 1;
%e 1, 6, 8, 6, 1;
%e 1, 7, 12, 12, 7, 1;
%e 1, 8, 17, 22, 17, 8, 1;
%e 1, 9, 23, 37, 37, 23, 9, 1;
%e 1, 10, 30, 58, 72, 58, 30, 10, 1;
%e 1, 11, 38, 86, 128, 128, 86, 38, 11, 1;
%e 1, 12, 47, 122, 212, 254, 212, 122, 47, 12, 1;
%e ...
%t T[n_, m_] = Binomial[n, m] + 2*If[m*(n - m) > 0, 1, 0];
%t Flatten[Table[T[n, m], {n, 0, 10}, {m, 0, n}]]
%o (Maxima) T(n,k) := if k = 0 or k = n then 1 else binomial(n, k) + 2$
%o create_list(T(n, k), n, 0, 12, k, 0, n); /* _Franck Maminirina Ramaharo_, Dec 08 2018 */
%o (Sage)
%o def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 2
%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 13 2021
%o (Magma)
%o T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) + 2 >;
%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 13 2021
%Y Cf. A100314, A103451, A131520, A156050.
%Y Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), A132735 (q=1), this sequence (q=2), A173741 (q=4), A173742 (q=6).
%K nonn,tabl,easy
%O 0,5
%A _Roger L. Bagula_, Feb 23 2010
%E Edited and name clarified by _Franck Maminirina Ramaharo_, Dec 08 2018
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