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%I #9 Mar 05 2022 01:00:54
%S 1,1,1,1,1,1,1,2,2,1,1,3,5,3,1,1,4,9,9,4,1,1,5,14,19,14,5,1,1,6,20,34,
%T 34,20,6,1,1,7,27,55,69,55,27,7,1,1,8,35,83,125,125,83,35,8,1,1,9,44,
%U 119,209,251,209,119,44,9,1
%N Triangle T(n,k) = binomial(n, k) - 1 with T(n,0) = T(n,n) = 1, read by rows.
%C Row sums = A132045: (1, 2, 3, 6, 13, 28, 59, ...).
%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 12 2021
%H G. C. Greubel, <a href="/A132044/b132044.txt">Rows n = 0..100 of the triangle, flattened</a>
%F T(n, k) = A007318(n,k) + A103451(n,k) - A000012(n,k), an infinite lower triangular matrix.
%F T(n, k) = binomial(n, k) - 1, with T(n,0) = T(n,n) = 1. - _Roger L. Bagula_, Feb 08 2010
%F From _G. C. Greubel_, Feb 12 2021: (Start)
%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 = 0.
%F Sum_{k=0..n} T(n, k, 0) = 2^n - (n-1) - [n=0]. (End)
%e First few rows of the triangle:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 2, 1;
%e 1, 3, 5, 3, 1;
%e 1, 4, 9, 9, 4, 1;
%e 1, 5, 14, 19, 14, 5, 1;
%e 1, 6, 20, 34, 34, 20, 6, 1;
%e 1, 7, 27, 55, 69, 55, 27, 7, 1;
%t T[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _Roger L. Bagula_, Feb 08 2010 *)
%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,0) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 12 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,0): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 12 2021
%Y Cf. A007318, A103451, A132045.
%Y Cf. this sequence (q=0), A173075 (q=1), A173046 (q=2), A173047 (q=3).
%K nonn,tabl
%O 0,8
%A _Gary W. Adamson_, Aug 08 2007
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