A002720 Comment and Asymptotic

[Alec Mihailovs]

"Franklin T. Adams-Watters" <franktaw at netscape.net> wrote:


> This leads to the question of the asymptotic behavior of this sequence. 
> There are two integral versions of the sequence b(n) in the OEIS: A065094 
> and A065095.  Each has a conjectured asymptotic of C*BesselI(0,2*sqrt(n)). 
> If this is correct, A002720 should be asymptotically 
> C*BesselI(0,2*sqrt(n))*n!.  This appears to be the case, with C 
> approximately equal to .6.  Convergence is slow, but C may be 6/Pi^2 
> (evaluating to 30,000, the value is .60857..., vs 6/Pi^2 = .60792...).

The asymptotic of A002720 can be found from its exponential generating 
function,

a(n)/n! ~ 1/2*1/Pi^(1/2)*exp(-1/2+2*n^(1/2))/n^(1/4)

Comparing it with

Bessel(0,2*sqrt(n)) ~ 1/2*1/Pi^(1/2)*exp(2*n^(1/2))/n^(1/4),

we get the conjectured asymptotic with C = exp(-1/2) = 
.6065306597126334236...

Alec Mihailovs
http://math.tntech.edu/alec/