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%I #16 Feb 14 2021 18:38:53
%S 1,1,1,1,2,1,1,4,4,1,1,6,10,6,1,1,8,18,18,8,1,1,10,28,38,28,10,1,1,12,
%T 40,68,68,40,12,1,1,14,54,110,138,110,54,14,1,1,16,70,166,250,250,166,
%U 70,16,1,1,18,88,238,418,502,418,238,88,18,1
%N Triangle T(n,k) = 2 * binomial(n,k) - 2 with T(n,0) = T(n,n) = 1, read by rows.
%H G. C. Greubel, <a href="/A132731/b132731.txt">Rows n = 0..100 of the triangle, flattened</a>
%F T(n, k) = 2*A007318 + A103451 - 2*A000012, an infinite lower triangular matrix.
%F From _G. C. Greubel_, Feb 14 2021: (Start)
%F T(n, k) = 2*binomial(n, k) - 2 with T(n, 0) = T(n, n) = 1.
%F T(n, k) = 2*A132044(n, k) with T(n, 0) = T(n, n) = 1.
%F Sum_{k=0..n} T(n, k) = 2^(n+1) - 2*n - [n=0] = A132732(n). (End)
%e First few rows of the triangle are:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 4, 4, 1;
%e 1, 6, 10, 6, 1;
%e 1, 8, 18, 18, 8, 1;
%e 1, 10, 28, 38, 28, 10, 1;
%e 1, 12, 40, 68, 68, 40, 12, 1;
%e ...
%t T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n, k] - 2];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 14 2021 *)
%o (PARI) t(n,k) = 2*binomial(n, k) + ((k==0) || (k==n)) - 2*(k<=n); \\ _Michel Marcus_, Feb 12 2014
%o (Sage)
%o def T(n, k): return 1 if (k==0 or k==n) else 2*binomial(n, k) - 2
%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 14 2021
%o (Magma)
%o T:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n,k) - 2 >;
%o [T(n,k): k in [0..n], n in [0..12]]; // __G. C. Greubel_, Feb 14 2021
%Y Cf. A000012, A007318, A103451, A132044, A132732 (row sums).
%K nonn,tabl
%O 0,5
%A _Gary W. Adamson_, Aug 26 2007
%E Corrected by _Jeremy Gardiner_, Feb 02 2014
%E More terms from _Michel Marcus_, Feb 12 2014
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