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A132737 Triangle T(n,k) = 2*binomial(n,k) + 1, read by rows.

%I A132737 #9 Feb 16 2021 01:05:29
%S A132737 1,1,1,1,5,1,1,7,7,1,1,9,13,9,1,1,11,21,21,11,1,1,13,31,41,31,13,1,1,
%T A132737 15,43,71,71,43,15,1,1,17,57,113,141,113,57,17,1,1,19,73,169,253,253,
%U A132737 169,73,19,1,1,21,91,241,421,505,421,241,91,21,1,1,23,111,331,661,925,925,661,331,111,23,1
%N A132737 Triangle T(n,k) = 2*binomial(n,k) + 1, read by rows.
%H A132737 G. C. Greubel, <a href="/A132737/b132737.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A132737 T(n, k) = 2*A132735(n, k) - 1, an infinite lower triangular matrix.
%F A132737 T(n,0) = T(n,n) = 1; otherwise T(n,k) = 2*C(n,k) + 1. - _Franklin T. Adams-Watters_, Jul 06 2009
%F A132737 Sum_{k=0..n} T(n, k) = 2^(n+1) + n - 3 + 2*[n=0] = A132738(n). - _G. C. Greubel_, Feb 15 2021
%e A132737 First few rows of the triangle are:
%e A132737   1;
%e A132737   1,  1;
%e A132737   1,  5,  1;
%e A132737   1,  7,  7,  1;
%e A132737   1,  9, 13,  9,  1;
%e A132737   1, 11, 21, 21, 11,  1;
%e A132737   1, 13, 31, 41, 31, 13,  1;
%e A132737   1, 15, 43, 71, 71, 43, 15, 1;
%e A132737   ...
%t A132737 T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n,k] +1];
%t A132737 Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 15 2021 *)
%o A132737 (Sage)
%o A132737 def A132737(n,k): return 1 if (k==0 or k==n) else 2*binomial(n,k) + 1
%o A132737 flatten([[A132737(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Feb 15 2021
%o A132737 (Magma)
%o A132737 A132737:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n,k) +1 >;
%o A132737 [A132737(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Feb 15 2021
%Y A132737 Cf. A132735, A132738.
%Y A132737 Sequences of the form 2*binomial(n,k) + q: A132729 (q=-3), A132731 (q=-2), A109128 (q=-1), A132046 (q=0), this sequence (q=1).
%K A132737 nonn,tabl
%O A132737 0,5
%A A132737 _Gary W. Adamson_, Aug 26 2007
%E A132737 Extended by _Franklin T. Adams-Watters_, Jul 06 2009