A112475 Riordan array (1/(1+x),x(1+x)/(1-x)).

1, -1, 1, 1, 1, 1, -1, 1, 3, 1, 1, 1, 5, 5, 1, -1, 1, 7, 13, 7, 1, 1, 1, 9, 25, 25, 9, 1, -1, 1, 11, 41, 63, 41, 11, 1, 1, 1, 13, 61, 129, 129, 61, 13, 1, -1, 1, 15, 85, 231, 321, 231, 85, 15, 1, 1, 1, 17, 113, 377, 681, 681, 377, 113, 17, 1

Offset: 0

Author

Paul Barry, Sep 07 2005

Keywords

• easy
• sign
• tabl

Comments

Equivalent to Delannoy triangle A008288 with prepended column 1,-1,1,-1,... Row sums are A111954. Diagonal sums are A112476. Inverse is A112477.

Examples

			Triangle starts:
   1;
  -1, 1;
   1, 1, 1;
  -1, 1, 3,  1;
   1, 1, 5,  5,  1;
  -1, 1, 7, 13,  7, 1;
   1, 1, 9, 25, 25, 9, 1;
  ...
		

Crossrefs

Cf. A008288, A111954, A112476, A112477.

Programs

Mathematica

T[n_,k_]:=SeriesCoefficient[(x(1+x)/(1-x))^k/(1+x),{x,0,n}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, May 26 2024 *)

Formula

T(n,k) = Sum{j=0..n-k} C(k-1,j)*C(n-j-1,n-k-j).