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