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Triangle T(n, k) = 2*A132749(n, k) - 1, read by rows.
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%I #5 Feb 16 2021 17:48:48

%S 1,3,1,3,3,1,3,5,5,1,3,7,11,7,1,3,9,19,19,9,1,3,11,29,39,29,11,1,3,13,

%T 41,69,69,41,13,1,3,15,55,111,139,111,55,15,1,3,17,71,167,251,251,167,

%U 71,17,1

%N Triangle T(n, k) = 2*A132749(n, k) - 1, read by rows.

%H G. C. Greubel, <a href="/A132752/b132752.txt">Rows n = 0..100 of the triangle, flattened</a>

%F T(n, k) = 2*A132749(n, k) - 1, an infinite lower triangular matrix.

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

%F T(n, k) = A109128(n, k) with T(n, 0) = 3.

%F Sum_{k=0..n} T(n, k) = 2^(n+1) -n +1 -2*[n=0] = A132753(n) - 2*[n=0]. (End)

%e First few rows of the triangle are:

%e 1;

%e 3, 1;

%e 3, 3, 1;

%e 3, 5, 5, 1;

%e 3, 7, 11, 7, 1;

%e 3, 9, 19, 19, 9, 1;

%e 3, 11, 29, 39, 29, 11, 1;

%e ...

%t T[n_, k_]:= If[k==n, 1, If[k==0, 3, 2*Binomial[n, k] -1 ]];

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

%o (Sage)

%o def A132752(n,k): return 1 if k==n else 3 if k==0 else 2*binomial(n,k) -1

%o flatten([[A132752(n,k) for k in [0..n]] for n in [0..12]]) # _G. C. Greubel_, Feb 16 2021

%o (Magma)

%o A132752:= func< n,k | k eq n select 1 else k eq 0 select 3 else 2*Binomial(n,k) -1 >;

%o [A132752(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 16 2021

%Y Cf. A109128, A132749, A132753.

%K nonn,tabl

%O 0,2

%A _Gary W. Adamson_, Aug 28 2007