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
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Jul 14 2010
STATUS
approved