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Triangle T(n,k) = binomial(n, k) with T(n, 0) = 2, read by rows.
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%I #16 Feb 16 2021 09:46:24

%S 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 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 45,120,210,252,210,120,45,10,1,2,11,55,165,330,462,462,330,165,55,11,1

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

%C Add 1 to all but the top entry in the left column of the Pascal matrix. - _R. J. Mathar_, Jan 18 2013

%H G. C. Greubel, <a href="/A132749/b132749.txt">Rows n = 0..100 of the triangle, flattened</a>

%F T(n,k) = A103451(n,k) * A007318(n,k), an infinite lower triangular matrix.

%F From _G. C. Greubel_, Feb 16 2021: (Start)

%F T(n,k) = binomial(n, k) with T(n, 0) = 2 for n>0.

%F Sum_{k=0..n} T(n, k) = A083318(n) = 2^n + 1^n - 0^n. (End)

%e First few rows of the triangle are:

%e 1;

%e 2, 1;

%e 2, 2, 1;

%e 2, 3, 3, 1;

%e 2, 4, 6, 4, 1;

%e 2, 5, 10, 10, 5, 1;

%e ...

%t T[n_, k_]:= If[k==n, 1, If[k==0, 2, Binomial[n, k]]];

%t Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Feb 16 2021 *)

%o (Sage)

%o def A132749(n,k): return 1 if k==n else 2 if k==0 else binomial(n,k)

%o flatten([[A132749(n,k) for k in [0..n]] for n in [0..12]]) # _G. C. Greubel_, Feb 16 2021

%o (Magma)

%o A132749:= func< n,k | k eq n select 1 else k eq 0 select 2 else Binomial(n,k) >;

%o [A132749(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 16 2021

%Y Cf. A007318, A083318 (row sums), A103451.

%K nonn,easy,tabl,less

%O 0,2

%A _Gary W. Adamson_, Aug 28 2007

%E More terms added by _G. C. Greubel_, Feb 16 2021