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

0,2

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).

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

Table of n, a(n) for n=0..11.

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.

Formula

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

Example

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.

Mathematica

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

Crossrefs

Cf. A060506, A014402, A014403.

Sequence in context: A111598 A116312 A111782 * A238772 A225831 A286930

Adjacent sequences: A060504 A060505 A060506 * A060508 A060509 A060510

Keyword

easy,frac,nonn

Author

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

Status

approved