A060506 - OEIS

%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