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

0,4

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

Links

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

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

Example

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;
...

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A007318, A095121, A097806.

Sequence in context: A308173 A256910 A181176 * A128255 A154948 A109091

Adjacent sequences: A131105 A131106 A131107 * A131109 A131110 A131111

Keyword

nonn,tabl

Author

Gary W. Adamson, Jun 15 2007

Extensions

Corrected by Philippe Deléham, Dec 17 2007

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

Status

approved