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

0,5

Links

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

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)

Example

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;
  ...

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A283796 A156580 A157528 * A128966 A055907 A259698

Adjacent sequences: A132728 A132729 A132730 * A132732 A132733 A132734

Keyword

nonn,tabl

Author

Gary W. Adamson, Aug 26 2007

Extensions

Corrected by Jeremy Gardiner, Feb 02 2014

More terms from Michel Marcus, Feb 12 2014

Status

approved