A132731 Triangle T(n,k) = 2 * binomial(n,k) - 2 with T(n,0) = T(n,n) = 1, read by rows.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 10, 6, 1, 1, 8, 18, 18, 8, 1, 1, 10, 28, 38, 28, 10, 1, 1, 12, 40, 68, 68, 40, 12, 1, 1, 14, 54, 110, 138, 110, 54, 14, 1, 1, 16, 70, 166, 250, 250, 166, 70, 16, 1, 1, 18, 88, 238, 418, 502, 418, 238, 88, 18, 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,  2,  1;
  1,  4,  4,  1;
  1,  6, 10,  6,  1;
  1,  8, 18, 18,  8,  1;
  1, 10, 28, 38, 28, 10,  1;
  1, 12, 40, 68, 68, 40, 12, 1;
  ...
		

Links

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

Crossrefs

Cf. A000012, A007318, A103451, A132044, A132732 (row sums).

Programs

Magma

T:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n,k) - 2 >;
[T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel, Feb 14 2021

Mathematica

T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n, k] - 2];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 14 2021 *)

PARI

t(n,k) =  2*binomial(n, k) + ((k==0) || (k==n)) - 2*(k<=n); \\ Michel Marcus, Feb 12 2014

Sage

def T(n, k): return 1 if (k==0 or k==n) else 2*binomial(n, k) - 2
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 14 2021

Formula

T(n, k) = 2*A007318 + A103451 - 2*A000012, an infinite lower triangular matrix.

From G. C. Greubel, Feb 14 2021: (Start)

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

T(n, k) = 2*A132044(n, k) with T(n, 0) = T(n, n) = 1.

Sum_{k=0..n} T(n, k) = 2^(n+1) - 2*n - [n=0] = A132732(n). (End)

Extensions

Corrected by Jeremy Gardiner, Feb 02 2014

More terms from Michel Marcus, Feb 12 2014