A132749 Triangle T(n,k) = binomial(n, k) with T(n, 0) = 2, read by rows.

1, 2, 1, 2, 2, 1, 2, 3, 3, 1, 2, 4, 6, 4, 1, 2, 5, 10, 10, 5, 1, 2, 6, 15, 20, 15, 6, 1, 2, 7, 21, 35, 35, 21, 7, 1, 2, 8, 28, 56, 70, 56, 28, 8, 1, 2, 9, 36, 84, 126, 126, 84, 36, 9, 1, 2, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 2, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 0

Author

Gary W. Adamson, Aug 28 2007

Keywords

• nonn
• easy
• tabl
• less

Comments

Add 1 to all but the top entry in the left column of the Pascal matrix. - R. J. Mathar, Jan 18 2013

Examples

			First few rows of the triangle are:
  1;
  2, 1;
  2, 2,  1;
  2, 3,  3,  1;
  2, 4,  6,  4, 1;
  2, 5, 10, 10, 5, 1;
  ...
		

Links

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

Crossrefs

Cf. A007318, A083318 (row sums), A103451.

Programs

Magma

A132749:= func< n,k | k eq n select 1 else k eq 0 select 2 else Binomial(n,k) >;
[A132749(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

Mathematica

T[n_, k_]:= If[k==n, 1, If[k==0, 2, Binomial[n, k]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 16 2021 *)

Sage

def A132749(n,k): return 1 if k==n else 2 if k==0 else binomial(n,k)
flatten([[A132749(n,k) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Feb 16 2021

Formula

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

From G. C. Greubel, Feb 16 2021: (Start)

T(n,k) = binomial(n, k) with T(n, 0) = 2 for n>0.

Sum_{k=0..n} T(n, k) = A083318(n) = 2^n + 1^n - 0^n. (End)

Extensions

More terms added by G. C. Greubel, Feb 16 2021