|
%I #10 Feb 15 2021 15:38:15
%S 1,1,1,1,3,1,1,4,4,1,1,5,7,5,1,1,6,11,11,6,1,1,7,16,21,16,7,1,1,8,22,
%T 36,36,22,8,1,1,9,29,57,71,57,29,9,1,1,10,37,85,127,127,85,37,10,1,1,
%U 11,46,121,211,253,211,121,46,11,1,1,12,56,166,331,463,463,331,166,56,12,1
%N Triangle T(n,k) = binomial(n,k) + 1 with T(n,0) = T(n,n) = 1, read by rows.
%H G. C. Greubel, <a href="/A132735/b132735.txt">Rows n = 0..100 of the triangle, flattened</a>
%F T(n, k) = A007318(n,k) + 1 - A103451(n,k), an infinite lower triangular matrix.
%F T(n,0) = T(n,n) = 1; T(n,k) = C(n,k) + 1 otherwise. - _Franklin T. Adams-Watters_, Jul 06 2009
%F Sum_{k=0..n} T(n, k) = 2^n + n - 1 + [n=0] = A132736(n). - _G. C. Greubel_, Feb 14 2021
%e First few rows of the triangle are:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 4, 4, 1;
%e 1, 5, 7, 5, 1;
%e 1, 6, 11, 11, 6, 1;
%e 1, 7, 16, 21, 16, 7, 1;
%e ...
%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 (* _G. C. Greubel_, Feb 14 2021 *)
%o (Sage)
%o def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 1
%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 Binomial(n,k) + 1 >;
%o [T(n,k): k in [0..n], n in [0..12]]; // __G. C. Greubel_, Feb 14 2021
%Y Cf. A103451, A132736.
%Y Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), this sequence (q=1), A173740 (q=2), A173741 (q=4), A173742 (q=6).
%K nonn,tabl
%O 0,5
%A _Gary W. Adamson_, Aug 26 2007
%E Corrected and extended by _Franklin T. Adams-Watters_, Jul 06 2009
|
