A087216 a(n) = (6n)!/((3n)!(2n)!2^n).

1, 30, 6930, 3063060, 2007835830, 1746817172100, 1895296631728500, 2464427134570401000, 3735455429225085315750, 6467318499798364376668500, 12591869119107415441373569500, 27232778758505946668207019855000

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

Sequence in context: A184889 A358481 A300147 * A239925 A059049 A056071

Adjacent sequences: A087213 A087214 A087215 * A087217 A087218 A087219

Keyword

nonn

Author

Michael Somos, Aug 26 2003

Status

approved