A119326 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,2j)*C(n-k,2j).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 7, 10, 7, 1, 1, 1, 1, 11, 19, 19, 11, 1, 1, 1, 1, 16, 31, 38, 31, 16, 1, 1, 1, 1, 22, 46, 66, 66, 46, 22, 1, 1, 1, 1, 29, 64, 106, 126, 106, 64, 29, 1, 1, 1, 1, 37, 85, 162, 226, 226, 162, 85, 37, 1, 1

Offset: 0

Author

Paul Barry, May 14 2006

Keywords

• easy
• nonn
• tabl

Comments

Third column is essentially A000124. Fourth column is essentially A005448. Fifth column is A119327. Product of Pascal's triangle A007318 and A119328. Row sums are A038504. T(n,k) = T(n,n-k).

Examples

			Triangle begins:
  1;
  1, 1;
  1, 1,  1;
  1, 1,  1,  1;
  1, 1,  2,  1,  1;
  1, 1,  4,  4,  1,  1;
  1, 1,  7, 10,  7,  1, 1;
  1, 1, 11, 19, 19, 11, 1, 1;
  ...
		

References

• Lukas Spiegelhofer and Jeffrey Shallit, Continuants, Run Lengths, and Barry's Modified Pascal Triangle, Volume 26(1) 2019, of The Electronic Journal of Combinatorics, #P1.31.

Links

• Seiichi Manyama, Rows n = 0..139, flattened
• Jeffrey Shallit, Lukas Spiegelhofer, Continuants, run lengths, and Barry's modified Pascal triangle, arXiv:1710.06203 [math.CO], 2017.

Crossrefs

Cf. A119358.

Formula

Column k has g.f.: (x^k/(1-x))* Sum{j=0..k} C(k,2j)*(x/(1-x))^(2j).

T(2n,n) = A119358(n). - Alois P. Heinz, Aug 31 2018