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A132729 Triangle T(n, k) = 2*binomial(n, k) - 3 with T(n, 0) = T(n, n) = 1, read by rows.

%I A132729 #11 Feb 13 2021 03:05:13
%S A132729 1,1,1,1,1,1,1,3,3,1,1,5,9,5,1,1,7,17,17,7,1,1,9,27,37,27,9,1,1,11,39,
%T A132729 67,67,39,11,1,1,13,53,109,137,109,53,13,1,1,15,69,165,249,249,165,69,
%U A132729 15,1,1,17,87,237,417,501,417,237,87,17,1,1,19,107,327,657,921,921,657,327,107,19,1
%N A132729 Triangle T(n, k) = 2*binomial(n, k) - 3 with T(n, 0) = T(n, n) = 1, read by rows.
%H A132729 G. C. Greubel, <a href="/A132729/b132729.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A132729 T(n, k) = 2*A132044(n, k) - 1.
%F A132729 From _G. C. Greubel_, Feb 13 2021: (Start)
%F A132729 T(n, k) = 2*binomial(n, k) - 3 with T(n, 0) = T(n, n) = 1.
%F A132729 Sum_{k=0..n} T(n, k) = 2^(n+1) - 3*n + 1 - 2*[n=0] = A132730(n). (End)
%e A132729 First few rows of the triangle are:
%e A132729   1;
%e A132729   1,  1;
%e A132729   1,  1,  1;
%e A132729   1,  3,  3,  1;
%e A132729   1,  5,  9,  5,  1;
%e A132729   1,  7, 17, 17,  7,  1;
%e A132729   1,  9, 27, 37, 26,  9,  1;
%e A132729   1, 11, 39, 67, 67, 39, 11, 1;
%t A132729 T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n, k] - 3];
%t A132729 Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Feb 13 2021 *)
%o A132729 (Sage)
%o A132729 def T(n,k): return 1 if (k==0 or k==n) else 2*binomial(n,k) - 3
%o A132729 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 13 2021
%o A132729 (Magma)
%o A132729 T:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n,k) - 3 >;
%o A132729 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 13 2021
%Y A132729 Cf. A007318, A132044, A132730.
%K A132729 nonn,tabl
%O A132729 0,8
%A A132729 _Gary W. Adamson_, Aug 26 2007
%E A132729 More terms added by _G. C. Greubel_, Feb 13 2021