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A176730 Denominators of coefficients of a series, called f, related to Airy functions. 3
1, 6, 180, 12960, 1710720, 359251200, 109930867200, 46170964224000, 25486372251648000, 17891433320656896000, 15565546988971499520000, 16437217620353903493120000, 20710894201645918401331200000, 30693545206839251070772838400000, 52854284846177190343870827724800000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The numerators are always 1.

f(z):=sum((1/a(n))*z^(3*n),n=0..infty) and g(z):=sum((1/b(n))*z^(3*n+1),n=0..infty) with b(n):=A176731(n) build the two independent Airy functions Ai(z) = c[1]*f(z) - c[2]*g(z) and Bi(z) =\sqrt(3)* (c[1]*f(z) + c[2]*g(z)) with c[1]:=1/(3^(2/3)*GAMMA(2/3)), approximately 0.35502805388781723926 (maple12 10 digits) and c[2]:=1/(3^(1/3)*GAMMA(1/3)), approximately 0.25881940379280679840.

If y := sum_{n=0..} x^(3*n)/a(n), then y'' = x*y. - Michael Somos, Jul 12 2019

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..200

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, 10.4.2 - 5. [alternative scanned copy].

W. Lang, The first 20 terms of the f(z) and g(z) functions.

NIST's Digital Library of Mathematical Functions, Airy and Related Functions (Maclaurin Series) by Frank W. J. Olver.

FORMULA

a(n) = denominator((3^n)*risefac(1/3,n)/(3*n)!) with the rising factorials risefac(k,n):=product(k+j,j=0..(n-1)) and risefac(k,0)=1.

EXAMPLE

Rational f-coefficients: [1, 1/6, 1/180, 1/12960, 1/1710720, 1/359251200, 1/109930867200, 1/46170964224000,...].

MATHEMATICA

a[ n_] := If[ n < 0, 0, 1 / (3^(2/3) Gamma[2/3] SeriesCoefficient[ AiryAi[x], {x, 0, 3 n}])]; (* Michael Somos, Oct 14 2011 *)

a[ n_] := If[ n < 0, 0, (3*n)! / Product[ k, {k, 1, 3 n - 2, 3}]]; (* Michael Somos, Oct 14 2011 *)

PROG

(PARI) {a(n) = if( n<0, 0, (3*n)! / prod( k=0, n-1, 3*k + 1))}; /* Michael Somos, Oct 14 2011 */

CROSSREFS

Cf. A176731.

Sequence in context: A210358 A135395 A141121 * A225776 A051357 A251671

Adjacent sequences:  A176727 A176728 A176729 * A176731 A176732 A176733

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Jul 14 2010

STATUS

approved

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Last modified October 14 12:29 EDT 2019. Contains 328006 sequences. (Running on oeis4.)