A060507 Denominators of the asymptotic expansion of the Airy function Ai(x).

1, 72, 3456, 746496, 214990848, 1719926784, 743008370688, 53496602689536, 10271347716390912, 6655833320221310976, 958439998111868780544, 23002559954684850733056

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((-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).

Examples

			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.
		

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. A060506, A014402, A014403.

Programs

Mathematica

a[ n_] := If[ n<0, 0, Denominator[ Product[k, {k, 1, 6 n - 1, 2}] / n! / 216^n]] (* Michael Somos, Oct 14 2011 *)

Formula

a(k)=denom(product((2*l+1), l=k..3*k-1)/216^k/k!).