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

0,8

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A007318, A132044, A132730.

Sequence in context: A056611 A177407 A135669 * A196493 A251634 A196989

Adjacent sequences: A132726 A132727 A132728 * A132730 A132731 A132732

Keyword

nonn,tabl

Author

Gary W. Adamson, Aug 26 2007

Extensions

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

Status

approved