A380652 Shifts left one place under the inverse modulo 2 binomial transform.

1, 1, 0, -1, -1, -2, -1, 1, 4, 3, -1, -4, -1, -3, 0, 5, 5, 4, -1, -5, -2, -5, -1, 6, 9, 1, -6, -5, 13, 14, 11, 1, -38, -39, -1, 38, 41, 81, 42, -37, -163, -128, 37, 167, 56, 143, 11, -214, -253, -219, 36, 257, 149, 328, 105, -303, -624, -247, 313, 490, -455, -387, -476, -417, 1251, 1250

Offset: 0

Author

Ilya Gutkovskiy, Jan 29 2025

Keywords

• sign

Crossrefs

Cf. A000587, A010060, A047999, A166966.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = Sum[(-1)^ThueMorse[k] Mod[Binomial[n - 1, k], 2] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 65}]

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def A380652(n): return sum((-A380652(n-k-1) if k.bit_count()&1 else A380652(n-k-1)) for k in range(n) if not (k&~(n-1))) if n else 1 # Chai Wah Wu, Feb 11 2025

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^A010060(k) * (binomial(n-1,k) mod 2) * a(n-k-1).