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

0,2

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_{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).

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

Example

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.

Mathematica

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

Crossrefs

Cf. A060507.

Sequence in context: A198902 A100474 A152438 * A302394 A057633 A193126

Adjacent sequences: A060503 A060504 A060505 * A060507 A060508 A060509

Keyword

easy,frac,nonn

Author

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

Status

approved