A307156 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,3*j) * binomial(n-k,3*j).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, -3, -3, 1, 1, 1, 1, 1, 1, -9, -15, -9, 1, 1, 1, 1, 1, 1, -19, -39, -39, -19, 1, 1, 1, 1, 1, 1, -34, -79, -99, -79, -34, 1, 1, 1, 1, 1, 1, -55, -139, -199, -199, -139, -55, 1, 1, 1

Offset

0,32

Links

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

Example

Triangle begins:
n\k | 0  1  2   3    4   5  6  7  8
----+-------------------------------
0   | 1;
1   | 1, 1;
2   | 1, 1, 1;
3   | 1, 1, 1,  1;
4   | 1, 1, 1,  1,   1;
5   | 1, 1, 1,  1,   1,  1;
6   | 1, 1, 1,  0,   1,  1, 1;
7   | 1, 1, 1, -3,  -3,  1, 1, 1;
8   | 1, 1, 1, -9, -15, -9, 1, 1, 1;

Mathematica

T[n_, k_] := Sum[(-1)^j * Binomial[k, 3*j] * Binomial[n - k, 3*j], {j, 0, n - k}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, May 20 2021 *)

Crossrefs

Row sums give A307089.

T(2*n,n) gives A307158.

Cf. A119326, A119335, A307090.

Sequence in context: A225054 A355604 A059790 * A326057 A130778 A353516

Adjacent sequences: A307153 A307154 A307155 * A307157 A307158 A307159

Keyword

sign,look,tabl

Author

Seiichi Manyama, Mar 27 2019

Status

approved