This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370070 #17 Mar 29 2025 03:27:04
%S A370070 0,1,2,4,10,38,274,5130,353186,180449810,1025875786562,
%T A370070 474164444389402658,13339869168335987186843266,
%U A370070 6036430661900479858398240235709517890,3241401154265052413102761158540183436937430482058498
%N A370070 a(n) = Sum_{i=0..n-1} binomial(2^i+2^(n-i-1)-2,2^i-1).
%F A370070 a(n) = A368548(2^n-1).
%F A370070 If n is odd, a(n) = binomial(2*(2^((n-1)/2)-1),2^((n-1)/2)-1) + 2*Sum_{i=0..(n-3)/2} binomial(2^i+2^(n-i-1)-2,2^i-1).
%F A370070 If n is even, a(n) = 2*Sum_{i=0..n/2-1} binomial(2^i+2^(n-i-1)-2,2^i-1).
%F A370070 log(a(n)) ~ c * 2^(n/2), where c = 3*log(3)/2 - log(2) if n is even and c = sqrt(2)*log(2) if n is odd. - _Vaclav Kotesovec_, Feb 10 2024
%t A370070 Table[Sum[Binomial[2^i+2^(n-i-1)-2,2^i-1],{i,0,n-1}],{n,0,14}] (* _James C. McMahon_, Feb 08 2024 *)
%o A370070 (Python)
%o A370070 from math import comb
%o A370070 def A370070(n): return (sum(comb((1<<i)+(1<<n-i-1)-2,(1<<i)-1) for i in range(n>>1))<<1) + (comb(((1<<(n>>1))-1)<<1,(1<<(n>>1))-1) if n&1 else 0)
%Y A370070 Cf. A368548.
%K A370070 nonn
%O A370070 0,3
%A A370070 _Chai Wah Wu_, Feb 08 2024