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

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, 15, 43, 71, 71, 43, 15, 1, 1, 17, 57, 113, 141, 113, 57, 17, 1, 1, 19, 73, 169, 253, 253, 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

Offset: 0

Author

Gary W. Adamson, Aug 26 2007

Keywords

• nonn
• tabl

Examples

			First few rows of the triangle are:
  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, 15, 43, 71, 71, 43, 15, 1;
  ...
		

Links

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

Crossrefs

Cf. A132735, A132738.

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).

Programs

Magma

A132737:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n,k) +1 >;
[A132737(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 15 2021

Mathematica

T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n,k] +1];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 15 2021 *)

Sage

def A132737(n,k): return 1 if (k==0 or k==n) else 2*binomial(n,k) + 1
flatten([[A132737(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Feb 15 2021

Formula

T(n, k) = 2*A132735(n, k) - 1, an infinite lower triangular matrix.

T(n,0) = T(n,n) = 1; otherwise T(n,k) = 2*C(n,k) + 1. - Franklin T. Adams-Watters, Jul 06 2009

Sum_{k=0..n} T(n, k) = 2^(n+1) + n - 3 + 2*[n=0] = A132738(n). - G. C. Greubel, Feb 15 2021

Extensions

Extended by Franklin T. Adams-Watters, Jul 06 2009