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A385087 2-adic valuation of A039699.

%I A385087 #20 Jun 18 2025 00:53:04
%S A385087 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,
%T A385087 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,
%U A385087 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
%N A385087 2-adic valuation of A039699.
%H A385087 Nikolai Beluhov, <a href="https://arxiv.org/abs/2506.12789">Powers of 2 in High-Dimensional Lattice Walks</a>, arXiv:2506.12789 [math.CO], 2025. See w4(n) in Table 1 p. 2.
%F A385087 a(n) = A007814(A039699(n)) = A000120(n) + A007814(A002895(n)).
%o A385087 (PARI) C=binomial;
%o A385087 A002895(n) = sum(k=0, n, C(n, k)^2 * C(2*n-2*k, n-k) * C(2*k, k) );
%o A385087 a(n) = hammingweight(n) + valuation(A002895(n), 2);
%o A385087 (Python)
%o A385087 from math import comb
%o A385087 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
%Y A385087 Cf. A000120, A002895, A007814, A039699.
%K A385087 nonn,easy
%O A385087 1,1
%A A385087 _Michel Marcus_, Jun 17 2025