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%I #5 Feb 16 2021 17:49:00
%S 1,1,1,1,5,1,1,10,10,1,1,19,37,19,1,1,36,105,105,36,1,1,69,270,403,
%T 270,69,1,1,134,660,1314,1314,660,134,1,1,263,1563,3895,5189,3895,
%U 1563,263,1,1,520,3619,10835,18045,18045,10835,3619,520,1,1,1033,8236,28791,57553,71931,57553,28791,8236,1033,1
%N Triangle T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 2, read by rows.
%C The triangle sequences having the form T(n,k,q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 have the row sums Sum_{k=0..n} T(n,k,q) = 2^(n-2)*q^n + 2^n - (n-1) - (5/4)*[n=0] -(q/2)*[n=1]. - _G. C. Greubel_, Feb 16 2021
%H G. C. Greubel, <a href="/A173046/b173046.txt">Rows n = 0..100 of the triangle, flattened</a>
%F T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 2.
%F Sum_{k=0..n} T(n, k, 2) = 4^(n-1) + 2^n - (n-1) - (5/4)*[n=0] = A000302(n-1) + A132045(n) - (5/4)*[n=0]. - [n=1]. - _G. C. Greubel_, Feb 16 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 5, 1;
%e 1, 10, 10, 1;
%e 1, 19, 37, 19, 1;
%e 1, 36, 105, 105, 36, 1;
%e 1, 69, 270, 403, 270, 69, 1;
%e 1, 134, 660, 1314, 1314, 660, 134, 1;
%e 1, 263, 1563, 3895, 5189, 3895, 1563, 263, 1;
%e 1, 520, 3619, 10835, 18045, 18045, 10835, 3619, 520, 1;
%e 1, 1033, 8236, 28791, 57553, 71931, 57553, 28791, 8236, 1033, 1;
%t T[n_, m_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] +(q^n)*Binomial[n-2, k-1] -1];
%t Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 16 2021 *)
%o (Sage)
%o def T(n,k,q): return 1 if (k==0 or k==n) else binomial(n,k) + q^n*binomial(n-2,k-1) -1
%o flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 16 2021
%o (Magma)
%o T:= func< n,k,q | k eq 0 or k eq n select 1 else Binomial(n,k) + q^n*Binomial(n-2,k-1) -1 >;
%o [T(n,k,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 16 2021
%Y Cf. A132044 (q=0), A173075 (q=1), this sequence (q=2), A173047 (q=3).
%Y Cf. A000302, A132045.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 08 2010
%E Edited by _G. C. Greubel_, Feb 16 2021
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