A119326 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,13

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).

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.

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

EXAMPLE

Triangle begins:

1, 1;

1, 1, 1;

1, 1, 1, 1;

1, 1, 2, 1, 1;