A002720 Comment and Asymptotic

[Franklin T. Adams-Watters]

I want to add the following comment for A002720:

Define b(n) by b(0) = 1, b(n) = b(n-1) + 1/n * Sum_{0<=k<n} b(k).  Then b(n) = a(n)/n!.

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...).

-- 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645


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