A134058 Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.

1, 2, 2, 2, 4, 2, 2, 6, 6, 2, 2, 8, 12, 8, 2, 2, 10, 20, 20, 10, 2, 2, 12, 30, 40, 30, 12, 2, 2, 14, 42, 70, 70, 42, 14, 2, 2, 16, 56, 112, 140, 112, 56, 16, 2, 2, 18, 72, 168, 252, 252, 168, 72, 18, 2

Offset: 0

Author

Gary W. Adamson, Oct 05 2007

Keywords

• nonn
• tabl

Comments

Triangle T(n,k), 0 <= k <= n, read by rows, given by [2, -1, 0, 0, 0, 0, 0, ...] DELTA [2, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 07 2007

Equals A028326 for all but the first term. - R. J. Mathar, Jun 08 2008

Warning: the row sums do not give A046055. - N. J. A. Sloane, Jul 08 2009

Examples

			First few rows of the triangle:
  1
  2,  2;
  2,  4,  2;
  2,  6,  6,  2;
  2,  8, 12,  8,  2;
  2, 10, 20, 20, 10,  2;
  ...
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A007318, A028326, A046055, A084938, A134059, A151821, A173048, A173049.

Programs

Magma

A134058:= func< n,k | n eq 0 select 1 else 2*Binomial(n,k) >;
[A134058(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 26 2021

Mathematica

T[n_, k_]:= SeriesCoefficient[(1+x+y)/(1-x-y), {x, 0, n-k}, {y, 0, k}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Apr 09 2015, after Vladimir Kruchinin *)
Table[2*Binomial[n,k] -Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 26 2021 *)

Sage

def A134058(n,k): return 2*binomial(n,k) - bool(n==0)
flatten([[A134058(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 26 2021

Formula

Double Pascal's triangle and replace leftmost column with (1,2,2,2,...).

M*A007318, where M = an infinite lower triangular matrix with (1,2,2,2,...) in the main diagonal and the rest zeros.

Sum_{k=0..n} T(n,k) = A151821(n+1). - Philippe Deléham, Sep 17 2009

G.f.: (1+x+y)/(1-x-y). - Vladimir Kruchinin, Apr 09 2015

T(n, k) = 2*binomial(n, k) - [n=0]. - G. C. Greubel, Apr 26 2021

E.g.f.: 2*exp(x*(1+y)) - 1. - Stefano Spezia, Apr 03 2024

Extensions

Title changed by G. C. Greubel, Apr 26 2021