|
%I #9 Feb 14 2021 18:39:09
%S 1,1,1,1,3,1,1,7,7,1,1,11,19,11,1,1,15,35,35,15,1,1,19,55,75,55,19,1,
%T 1,23,79,135,135,79,23,1,1,27,107,219,275,219,107,27,1,1,31,139,331,
%U 499,499,331,139,31,1,1,35,175,475,835,1003,835,475,175,35,1
%N Triangle T(n, k) = 4*binomial(n, k) - 5 with T(n, 0) = T(n, n) = 1, read by rows.
%H G. C. Greubel, <a href="/A132733/b132733.txt">Rows n = 0..100 of the triangle, flattened</a>
%F T(n, k) = 2*A132731 - A000012, an infinite lower triangular matrix.
%F From _G. C. Greubel_, Feb 14 2021: (Start)
%F T(n, k) = 4*binomial(n, k) - 5 with T(n, 0) = T(n, n) = 1.
%F Sum_{k=0..n} T(n, k) = 2^(n + 2) - (5*n + 1) - 2*[n=0] = A132734(n). (End)
%e First few rows of the triangle are:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 7, 7, 1;
%e 1, 11, 19, 11, 1;
%e 1, 15, 35, 35, 15, 1;
%e 1, 19, 55, 75, 55, 19, 1;
%e ...
%t T[n_, k_]:= If[k==0 || k==n, 1, 4*Binomial[n, k] - 5];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 14 2021 *)
%o (PARI) t(n,k) = 4*binomial(n, k) + 2*((k==0) || (k==n)) - 5*(k<=n); \\ _Michel Marcus_, Feb 12 2014
%o (Sage)
%o def T(n, k): return 1 if (k==0 or k==n) else 4*binomial(n, k) - 5
%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 4*Binomial(n,k) - 5 >;
%o [T(n,k): k in [0..n], n in [0..12]]; // __G. C. Greubel_, Feb 14 2021
%Y Cf. A132731, A132734.
%K nonn,tabl
%O 0,5
%A _Gary W. Adamson_, Aug 26 2007
%E More terms from _Michel Marcus_, Feb 12 2014
|
