Asymptotic for A058797

[Alec Mihailovs]

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


> The entry for A058797 states that it is asymptotic to c*n!, with c=.224...
> I conjecture that c = BesselJ(0,2) = Sum (-1)^k/(k!)^2 = 0.223890779... 
> (A091681).

I found a formula for A007754:

a(n,k) = Pi*(BesselJ(n+k+1,2)*BesselY(k,2) - BesselY(n+k+1,2)*BesselJ(k,2))

In particular, for k=0, we get a formula for A058797,

a(n) = Pi (BesselJ(n + 1, 2)*BesselY(0, 2) - BesselY(n + 1, 2) BesselJ(0, 
2))

Now, BesselJ(n+1,2) ~ 1/(n+1)! and BesselY(n+1,2) ~ -n!/Pi

That gives exactly that asymptotics, a(n) ~ BesselJ(0,2)*n!

Similarly, the asymptotics for the k-th column of A007754 is

a(n,k) ~ BesselJ(k,2)*(n+k)!

(for fixed k and n -> infinity).

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