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

0,5

Links

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

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

Example

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;
  ...

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 *)

Prog

(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
(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

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).

Sequence in context: A111720 A029646 A152721 * A137754 A131404 A297943

Adjacent sequences: A132734 A132735 A132736 * A132738 A132739 A132740

Keyword

nonn,tabl

Author

Gary W. Adamson, Aug 26 2007

Extensions

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

Status

approved