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A132749 Triangle T(n,k) = binomial(n, k) with T(n, 0) = 2, read by rows.

%I A132749 #16 Feb 16 2021 09:46:24
%S A132749 1,2,1,2,2,1,2,3,3,1,2,4,6,4,1,2,5,10,10,5,1,2,6,15,20,15,6,1,2,7,21,
%T A132749 35,35,21,7,1,2,8,28,56,70,56,28,8,1,2,9,36,84,126,126,84,36,9,1,2,10,
%U A132749 45,120,210,252,210,120,45,10,1,2,11,55,165,330,462,462,330,165,55,11,1
%N A132749 Triangle T(n,k) = binomial(n, k) with T(n, 0) = 2, read by rows.
%C A132749 Add 1 to all but the top entry in the left column of the Pascal matrix. - _R. J. Mathar_, Jan 18 2013
%H A132749 G. C. Greubel, <a href="/A132749/b132749.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A132749 T(n,k) = A103451(n,k) * A007318(n,k), an infinite lower triangular matrix.
%F A132749 From _G. C. Greubel_, Feb 16 2021: (Start)
%F A132749 T(n,k) = binomial(n, k) with T(n, 0) = 2 for n>0.
%F A132749 Sum_{k=0..n} T(n, k) = A083318(n) = 2^n + 1^n - 0^n. (End)
%e A132749 First few rows of the triangle are:
%e A132749   1;
%e A132749   2, 1;
%e A132749   2, 2,  1;
%e A132749   2, 3,  3,  1;
%e A132749   2, 4,  6,  4, 1;
%e A132749   2, 5, 10, 10, 5, 1;
%e A132749   ...
%t A132749 T[n_, k_]:= If[k==n, 1, If[k==0, 2, Binomial[n, k]]];
%t A132749 Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Feb 16 2021 *)
%o A132749 (Sage)
%o A132749 def A132749(n,k): return 1 if k==n else 2 if k==0 else binomial(n,k)
%o A132749 flatten([[A132749(n,k) for k in [0..n]] for n in [0..12]]) # _G. C. Greubel_, Feb 16 2021
%o A132749 (Magma)
%o A132749 A132749:= func< n,k | k eq n select 1 else k eq 0 select 2 else Binomial(n,k) >;
%o A132749 [A132749(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 16 2021
%Y A132749 Cf. A007318, A083318 (row sums), A103451.
%K A132749 nonn,easy,tabl,less
%O A132749 0,2
%A A132749 _Gary W. Adamson_, Aug 28 2007
%E A132749 More terms added by _G. C. Greubel_, Feb 16 2021