A119335 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,3j)*C(n-k,3j).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 11, 17, 11, 1, 1, 1, 1, 1, 1, 21, 41, 41, 21, 1, 1, 1, 1, 1, 1, 36, 81, 101, 81, 36, 1, 1, 1, 1, 1, 1, 57, 141, 201, 201, 141, 57, 1, 1, 1

Offset: 0

Author

Paul Barry, May 14 2006

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A119336. Product of Pascal's triangle and A119337.

Examples

			Triangle begins
1;
1, 1;
1, 1, 1;
1, 1, 1,  1;
1, 1, 1,  1,  1;
1, 1, 1,  1,  1,   1;
1, 1, 1,  2,  1,   1,  1;
1, 1, 1,  5,  5,   1,  1,  1;
1, 1, 1, 11, 17,  11,  1,  1, 1;
1, 1, 1, 21, 41,  41, 21,  1, 1, 1;
1, 1, 1, 36, 81, 101, 81, 36, 1, 1, 1;
		

Links

• Seiichi Manyama, Rows n = 0..139, flattened

Crossrefs

T(2n,n) gives A119363.

Cf. A119326.

Programs

Mathematica

T[n_, k_] := Sum[Binomial[k, 3j] Binomial[n-k, 3j], {j, 0, n-k}];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 14 2023 *)

Formula

Column k has g.f. (x^k/(1-x)) * Sum_{j=0..k} C(k,3j)(x/(1-x))^(3j).

Extensions

More terms from Seiichi Manyama, Mar 12 2019