A176731 Denominators of coefficients of a series, called g, related to Airy functions.

1, 12, 504, 45360, 7076160, 1698278400, 580811212800, 268334780313600, 161000868188160000, 121716656350248960000, 113196490405731532800000, 127006462235230779801600000, 169172607697327398695731200000, 263909268007830741965340672000000, 476620138022142319989405253632000000

Offset

0,2

Comments

The numerators are always 1.

f(z) := Sum_{n>=0} (1/b(n)) * z^(3*n) with b(n) := A176730(n) and g(z) := Sum_{n>=0} (1/a(n)) * z^(3*n+1) 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, and c(2) := 1/(3^(1/3) * Gamma(1/3)), approximately 0.25881940379280679840.

If y := Sum_{n >= 0} x^(3*n+1)/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].

Wolfdieter 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(2/3, n)/(3*n + 1)!) with the rising factorials risefac(k, n) := Product_{j=0..(n-1)} (k+j) and risefac(k, 0) = 1.

From Peter Bala, Dec 17 2021: (Start)

a(n) = 3*n*(3*n + 1)*a(n-1) with a(0) = 1.

a(n) = (3*n + 2)!/(n!*3^n)*Sum_{k = 0..n} (-1)^k*binomial(n,k)/(3*k + 2).

a(n) = (1/2)*(3*n + 2)!/(n!*3^n)*hypergeom([-n, 2/3], [5/3], 1).

a(n) = (2*Pi*sqrt(3))/9 *(1/3^(n+1))*Gamma(3*n+4)/( (n+1)*Gamma(1/3)* Gamma(n + 5/3) ). (End)

a(n) = (9^n*n!*(n + 1/3)!)/(1/3)!. - Peter Luschny, Dec 20 2021

Example

Rational g-coefficients: [1, 1/12, 1/504, 1/45360, 1/7076160, 1/1698278400, 1/580811212800, 1/268334780313600, ...].

Maple

a := proc (n) option remember; if n = 0 then 1 else 3*n*(3*n+1)*a(n-1) end if; end proc: seq(a(n), n = 0..20); # - Peter Bala, Dec 17 2021

Mathematica

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

Prog

(PARI) {a(n) = if( n<0, 0, (3*n + 1)! / prod( k=0, n-1, 3*k + 2))}; /* Michael Somos, Oct 14 2011 */

Crossrefs

Cf. A176730.

Sequence in context: A220322 A279302 A227052 * A211085 A197600 A197984

Adjacent sequences: A176728 A176729 A176730 * A176732 A176733 A176734

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Jul 14 2010

Status

approved