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

0,2

Comments

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

Links

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

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)

Example

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;
  ...

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A070080 A230196 A131400 * A271106 A273673 A376570

Adjacent sequences: A132746 A132747 A132748 * A132750 A132751 A132752

Keyword

nonn,easy,tabl,less

Author

Gary W. Adamson, Aug 28 2007

Extensions

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

Status

approved