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
Examples
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;
Links
- Seiichi Manyama, Rows n = 0..139, flattened
Crossrefs
Programs
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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 *)