A131108 T(n,k) = 2*A007318(n,k) - A097806(n,k).

1, 1, 1, 2, 3, 1, 2, 6, 5, 1, 2, 8, 12, 7, 1, 2, 10, 20, 20, 9, 1, 2, 12, 30, 40, 30, 11, 1, 2, 14, 42, 70, 70, 42, 13, 1, 2, 16, 56, 112, 140, 112, 56, 15, 1, 2, 18, 72, 168, 252, 252, 168, 72, 17, 1, 2, 20, 90, 240, 420, 504, 420, 240, 90, 19, 1

Offset: 0

Author

Gary W. Adamson, Jun 15 2007

Keywords

• nonn
• tabl

Comments

Row sums give A095121.

Triangle T(n,k), 0 <= k <= n, read by rows given by [1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 18 2007

Examples

			First few rows of the triangle are:
  1;
  1,  1;
  2,  3,  1;
  2,  6,  5,  1;
  2,  8, 12,  7, 1;
  2, 10, 20, 20, 9, 1;
...
		

Links

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

Crossrefs

Cf. A007318, A095121, A097806.

Programs

Magma

function T(n,k)
  if k eq n-1 then return 2*n-1;
  elif k eq n then return 1;
  else return 2*Binomial(n,k);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 18 2019

Maple

seq(seq( `if`(k=n-1, 2*n-1, `if`(k=n, 1, 2*binomial(n,k))), k=0..n), n=0..12); # G. C. Greubel, Nov 18 2019

Mathematica

Table[If[k==n-1, 2*n-1, If[k==n, 1, 2*Binomial[n, k]]], {n,0,12}, {k,0, n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

PARI

T(n,k) = if(k==n-1, 2*n-1, if(k==n, 1, 2*binomial(n,k))); \\ G. C. Greubel, Nov 18 2019

Sage

@CachedFunction
def T(n, k):
    if (k==n-1): return 2*n-1
    elif (k==n): return 1
    else: return 2*binomial(n,k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 18 2019

Formula

Twice Pascal's triangle minus A097806, the pairwise operator.

G.f.: (1-x*y+x^2+x^2*y)/((-1+x+x*y)*(x*y-1)). - R. J. Mathar, Aug 11 2015

Extensions

Corrected by Philippe Deléham, Dec 17 2007

More terms added and data corrected by G. C. Greubel, Nov 18 2019