A131240 T(n,k) = 2*A046854(n,k) - I.

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

Offset: 0

Author

Gary W. Adamson, Jun 21 2007

Keywords

• nonn
• tabl

Comments

Row sums = A001595: (1, 3, 5, 9, 15, 25, 41, 67, ...).

A131241 = 3*A046854 - 2*I.

Examples

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

Links

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

Crossrefs

Cf. A001595, A046854, A131241.

Programs

GAP

T:= function(n,k)
    if k=n then return 1;
    else return 2*Binomial(Int((n+k)/2), k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k)))); # G. C. Greubel, Jul 12 2019

Magma

[k eq n select 1 else 2*Binomial(Floor((n+k)/2), k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2019

Mathematica

Table[If[k==n, 1, 2*Binomial[Floor[(n+k)/2], k]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 12 2019 *)

PARI

T(n,k) = if(k==n, 1, 2*binomial((n+k)\2, k));

Sage

def T(n, k):
    if (k==n): return 1
    else: return 2*binomial(floor((n+k)/2), k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 12 2019

Formula

T(n,k) = 2*A046854(n,k) - Identity matrix, where A046854 = Pascal's triangle with repeats by columns.

Extensions

More terms added by G. C. Greubel, Jul 12 2019