A060506 Numerators of the asymptotic expansion of the Airy function Ai(x).

1, 5, 385, 425425, 1301375075, 188699385875, 2252127170418125, 6344885703973691875, 64115070038654156396875, 2830616227136542350765634375, 34904328696820703727291037478125, 88069967543659875631905704109578125

Offset: 0

Author

Michael Praehofer (praehofer(AT)ma.tum.de), Mar 22 2001

Keywords

• easy
• frac
• nonn

Comments

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).

Examples

			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.
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

Links

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• NIST's Digital Library of Mathematical Functions, Airy and Related Functions (Poincaré-Type Expansions) by Frank W. J. Olver.

Crossrefs

Cf. A060507.

Programs

Mathematica

a[ n_] := Numerator[Product[k, {k, 1, 6 n - 1, 2}] / n! / 216^n] (* Michael Somos, Oct 14 2011 *)

Formula

a(n) = numerator((Product_{k=1..3*n-1} (2*k+1))/(216^n*n!)). [Corrected by Sean A. Irvine, Nov 26 2022]