A370070 a(n) = Sum_{i=0..n-1} binomial(2^i+2^(n-i-1)-2,2^i-1).

0, 1, 2, 4, 10, 38, 274, 5130, 353186, 180449810, 1025875786562, 474164444389402658, 13339869168335987186843266, 6036430661900479858398240235709517890, 3241401154265052413102761158540183436937430482058498

Offset: 0

Author

Chai Wah Wu, Feb 08 2024

Keywords

• nonn

Crossrefs

Cf. A368548.

Programs

Mathematica

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 *)

Python

from math import comb
def A370070(n): return (sum(comb((1<>1))<<1) + (comb(((1<<(n>>1))-1)<<1,(1<<(n>>1))-1) if n&1 else 0)

Formula

a(n) = A368548(2^n-1).

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).

If n is even, a(n) = 2*Sum_{i=0..n/2-1} binomial(2^i+2^(n-i-1)-2,2^i-1).

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