OFFSET
0,2
COMMENTS
G.f. A(x) yields asymptotic expansion of Airy function Ai(x) ~ f((2/3) x^(3/2)) / (2 sqrt(pi) x^(1/4)) where f(x) = A(-1 / (432 x)) / exp(x).
G.f. A(x) yields asymptotic expansion of Airy function Bi(x) ~ f((2/3) x^(3/2)) / (sqrt(pi) x^(1/4)) where f(x) = A(1 / (432 x)) * exp(x).
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 448.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..230
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].
F. W. J. Olver, Link to a section of the Digital Library of Mathematical Functions.
R. P. Stanley, Recent Progress in Algebraic Combinatorics, Bull. Amer. Math. Soc., 40 (2003), 55-68.
FORMULA
G.f. A(x) satisfies 216 * x^2 * A(x)'' + (432 * x - 1) * A(x)' + 30 * A(x) = 0.
n*a(n) -6*(6*n-1)*(6*n-5)*a(n-1)=0. - R. J. Mathar, Feb 21 2013
G.f: 2F0(1/6,5/6;;216*x). - Benedict W. J. Irwin, Jul 13 2016
a(n) ~ 2^(3*n-1/2)*27^n*n^(n-1/2)*exp(-n)/sqrt(Pi). - Ilya Gutkovskiy, Jul 13 2016
MATHEMATICA
a[ n_] := If[ n < 0, 0, (6 n)! / ((3 n)! (2 n)! 2^n)]
CoefficientList[Series[HypergeometricPFQ[{1/6, 5/6}, {}, 216*x], {x, 0, 10}], x] (* Benedict W. J. Irwin, Jul 13 2016 *)
PROG
(PARI) {a(n) = if( n<0, 0, (6*n)! / (3*n)! / (2*n)! / 2^n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 26 2003
STATUS
approved