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%I #14 May 08 2018 15:11:55
%S 1,72,3456,746496,214990848,1719926784,743008370688,53496602689536,
%T 10271347716390912,6655833320221310976,958439998111868780544,
%U 23002559954684850733056
%N Denominators of the asymptotic expansion of the Airy function Ai(x).
%C The series arises in the asymptotic expansion of the Airy function A(x) for large |x| as Ai(x)~pi^(-1/2)/2*x^(-1/4)*exp(-z)*sum((-1)^k*c(k)*z^(-k),k=0..infinity), where z=2/3*x^(3/2). a(k) is the denominator of the fully canceled c(k).
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H NIST's Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/9.7#ii">Airy and Related Functions (Poincaré-Type Expansions)</a> by Frank W. J. Olver.
%F a(k)=denom(product((2*l+1), l=k..3*k-1)/216^k/k!).
%e a(2) = 3456 because for k=2, product((2*l+1),l=k..3*k-1)/216^k/k! = 385/3456 and we take the denominator of the fully canceled fraction.
%t a[ n_] := If[ n<0, 0, Denominator[ Product[k, {k, 1, 6 n - 1, 2}] / n! / 216^n]] (* _Michael Somos_, Oct 14 2011 *)
%Y Cf. A060506, A014402, A014403.
%K easy,frac,nonn
%O 0,2
%A Michael Praehofer (praehofer(AT)ma.tum.de), Mar 22 2001
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