A132044 Triangle T(n,k) = binomial(n, k) - 1 with T(n,0) = T(n,n) = 1, read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 1, 1, 4, 9, 9, 4, 1, 1, 5, 14, 19, 14, 5, 1, 1, 6, 20, 34, 34, 20, 6, 1, 1, 7, 27, 55, 69, 55, 27, 7, 1, 1, 8, 35, 83, 125, 125, 83, 35, 8, 1, 1, 9, 44, 119, 209, 251, 209, 119, 44, 9, 1

Offset

0,8

Comments

Row sums = A132045: (1, 2, 3, 6, 13, 28, 59, ...).

The triangle sequences having the form T(n,k,q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 have the row sums Sum_{k=0..n} T(n,k,q) = 2^(n-2)*q^n + 2^n - (n-1) - (5/4)*[n=0] -(q/2)*[n=1]. - G. C. Greubel, Feb 12 2021

Links

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

Formula

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

T(n, k) = binomial(n, k) - 1, with T(n,0) = T(n,n) = 1. - Roger L. Bagula, Feb 08 2010

From G. C. Greubel, Feb 12 2021: (Start)

T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 0.

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

Example

First few rows of the triangle:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,  1;
  1, 3,  5,  3,  1;
  1, 4,  9,  9,  4,  1;
  1, 5, 14, 19, 14,  5,  1;
  1, 6, 20, 34, 34, 20,  6, 1;
  1, 7, 27, 55, 69, 55, 27, 7, 1;

Mathematica

T[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Roger L. Bagula, Feb 08 2010 *)

Prog

(Sage)
def T(n, k, q): return 1 if (k==0 or k==n) else binomial(n, k) + q^n*binomial(n-2, k-1) -1
flatten([[T(n, k, 0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 12 2021
(Magma)
T:= func< n, k, q | k eq 0 or k eq n select 1 else Binomial(n, k) + q^n*Binomial(n-2, k-1) -1 >;
[T(n, k, 0): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 12 2021

Crossrefs

Cf. A007318, A103451, A132045.

Cf. this sequence (q=0), A173075 (q=1), A173046 (q=2), A173047 (q=3).

Sequence in context: A161671 A144444 A054106 * A034327 A034254 A157103

Adjacent sequences: A132041 A132042 A132043 * A132045 A132046 A132047

Keyword

nonn,tabl

Author

Gary W. Adamson, Aug 08 2007

Status

approved