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%I #24 Nov 26 2022 16:56:11
%S 1,5,385,425425,1301375075,188699385875,2252127170418125,
%T 6344885703973691875,64115070038654156396875,
%U 2830616227136542350765634375,34904328696820703727291037478125,88069967543659875631905704109578125
%N Numerators 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_{k>=0} (-1)^k*c(k)*z^(-k)), where z = (2/3)*x^(3/2). a(k) is the numerator 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(n) = numerator((Product_{k=1..3*n-1} (2*k+1))/(216^n*n!)). [Corrected by _Sean A. Irvine_, Nov 26 2022]
%e a(2)=385 because for n=2, (Product_{k=1..3*n-1} (2*k+1))/(216^n*n!) = 385/3456 and we take the numerator of the fully canceled fraction.
%t a[ n_] := Numerator[Product[k, {k, 1, 6 n - 1, 2}] / n! / 216^n] (* _Michael Somos_, Oct 14 2011 *)
%Y Cf. A060507.
%K easy,frac,nonn
%O 0,2
%A Michael Praehofer (praehofer(AT)ma.tum.de), Mar 22 2001
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