A132729 Triangle T(n, k) = 2*binomial(n, k) - 3 with T(n, 0) = T(n, n) = 1, read by rows.

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 9, 5, 1, 1, 7, 17, 17, 7, 1, 1, 9, 27, 37, 27, 9, 1, 1, 11, 39, 67, 67, 39, 11, 1, 1, 13, 53, 109, 137, 109, 53, 13, 1, 1, 15, 69, 165, 249, 249, 165, 69, 15, 1, 1, 17, 87, 237, 417, 501, 417, 237, 87, 17, 1, 1, 19, 107, 327, 657, 921, 921, 657, 327, 107, 19, 1

Offset: 0

Author

Gary W. Adamson, Aug 26 2007

Keywords

• nonn
• tabl

Examples

			First few rows of the triangle are:
  1;
  1,  1;
  1,  1,  1;
  1,  3,  3,  1;
  1,  5,  9,  5,  1;
  1,  7, 17, 17,  7,  1;
  1,  9, 27, 37, 26,  9,  1;
  1, 11, 39, 67, 67, 39, 11, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Crossrefs

Cf. A007318, A132044, A132730.

Programs

Magma

T:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n,k) - 3 >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021

Mathematica

T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n, k] - 3];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 13 2021 *)

Sage

def T(n,k): return 1 if (k==0 or k==n) else 2*binomial(n,k) - 3
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021

Formula

T(n, k) = 2*A132044(n, k) - 1.

From G. C. Greubel, Feb 13 2021: (Start)

T(n, k) = 2*binomial(n, k) - 3 with T(n, 0) = T(n, n) = 1.

Sum_{k=0..n} T(n, k) = 2^(n+1) - 3*n + 1 - 2*[n=0] = A132730(n). (End)

Extensions

More terms added by G. C. Greubel, Feb 13 2021