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A087216 a(n) = (6n)!/((3n)!(2n)!2^n). 1
1, 30, 6930, 3063060, 2007835830, 1746817172100, 1895296631728500, 2464427134570401000, 3735455429225085315750, 6467318499798364376668500, 12591869119107415441373569500, 27232778758505946668207019855000 (list; graph; refs; listen; history; text; internal format)
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
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].
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
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 26 2003
STATUS
approved

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Last modified July 15 21:59 EDT 2024. Contains 374334 sequences. (Running on oeis4.)