A385087 2-adic valuation of A039699.

3, 3, 10, 3, 6, 8, 12, 3, 6, 6, 11, 8, 11, 12, 16, 3, 6, 6, 12, 6, 9, 13, 17, 8, 11, 11, 16, 12, 15, 16, 20, 3, 6, 6, 14, 6, 9, 11, 15, 6, 9, 9, 14, 13, 16, 17, 21, 8, 11, 11, 17, 11, 14, 17, 21, 12, 15, 15, 20, 16, 19, 20, 24, 3, 6, 6, 13, 6, 9, 11, 15, 6, 9, 9, 14, 11, 14, 15, 19, 6

Offset: 1

Author

Michel Marcus, Jun 17 2025

Keywords

• nonn
• easy

Links

• Nikolai Beluhov, Powers of 2 in High-Dimensional Lattice Walks, arXiv:2506.12789 [math.CO], 2025. See w4(n) in Table 1 p. 2.

Crossrefs

Cf. A000120, A002895, A007814, A039699.

Programs

PARI

C=binomial;
A002895(n) = sum(k=0, n, C(n, k)^2 * C(2*n-2*k, n-k) * C(2*k, k) );
a(n) = hammingweight(n) + valuation(A002895(n), 2);

Python

from math import comb
def A385087(n): return (~(a:=(sum(comb(n,k)**2*comb(n-k<<1,n-k)*comb(m:=k<<1,k) for k in range(n+1>>1))<<1) + (0 if n&1 else comb(n,n>>1)**4)) & a-1).bit_length() + n.bit_count() # Chai Wah Wu, Jun 17 2025

Formula

a(n) = A007814(A039699(n)) = A000120(n) + A007814(A002895(n)).