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A060506
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Numerators of the asymptotic expansion of the Airy function Ai(x).
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2
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1, 5, 385, 425425, 1301375075, 188699385875, 2252127170418125, 6344885703973691875, 64115070038654156396875, 2830616227136542350765634375, 34904328696820703727291037478125, 88069967543659875631905704109578125
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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 numerator of the fully canceled c(k).
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).
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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].
NIST's Digital Library of Mathematical Functions, Airy and Related Functions (Poincare-Type Expansions) by Frank W. J. Olver.
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FORMULA
| a(k)=numer(product((2*l+1), l=k..3*k-1)/216^k/k!)
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EXAMPLE
| a(2)=385 because for k=2, product((2*l+1),l=k..3*k-1)/216^k/k! = 385/3456 and we take the numerator of the fully canceled fraction.
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MATHEMATICA
| a[ n_] := Numerator[Product[k, {k, 1, 6 n - 1, 2}] /n! / 216^n] (* Michael Somos, Oct 14 2011 *)
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CROSSREFS
| Cf. A060507.
Sequence in context: A198902 A100474 A152438 * A057633 A193126 A006700
Adjacent sequences: A060503 A060504 A060505 * A060507 A060508 A060509
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KEYWORD
| easy,frac,nonn
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AUTHOR
| Michael Praehofer (praehofer(AT)ma.tum.de), Mar 22 2001
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