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A176731
Denominators of coefficients of a series, called g, related to Airy functions.
3
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
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].
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
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Jul 14 2010
STATUS
approved