A307090 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,2*j) * binomial(n-k,2*j).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, -2, -2, 1, 1, 1, 1, -5, -8, -5, 1, 1, 1, 1, -9, -17, -17, -9, 1, 1, 1, 1, -14, -29, -34, -29, -14, 1, 1, 1, 1, -20, -44, -54, -54, -44, -20, 1, 1, 1, 1, -27, -62, -74, -74, -74, -62, -27, 1, 1, 1, 1, -35, -83, -90, -74, -74, -90, -83, -35, 1, 1

Offset

0,18

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,   0,   1,   1;
5   | 1, 1,  -2,  -2,   1,   1;
6   | 1, 1,  -5,  -8,  -5,   1,   1;
7   | 1, 1,  -9, -17, -17,  -9,   1, 1;
8   | 1, 1, -14, -29, -34, -29, -14, 1, 1;

Mathematica

T[n_, k_] := Sum[(-1)^j * Binomial[k, 2*j] * Binomial[n - k, 2*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 A099587(n+1).

T(2*n,n) gives A307091.

Cf. A007318, A098593, A119326, A119335.

Sequence in context: A096601 A078077 A078082 * A347630 A079674 A113193

Adjacent sequences: A307087 A307088 A307089 * A307091 A307092 A307093

Keyword

sign,tabl,look

Author

Seiichi Manyama, Mar 24 2019

Status

approved