A176730 Denominators of coefficients of a series, called f, related to Airy functions.

1, 6, 180, 12960, 1710720, 359251200, 109930867200, 46170964224000, 25486372251648000, 17891433320656896000, 15565546988971499520000, 16437217620353903493120000, 20710894201645918401331200000, 30693545206839251070772838400000, 52854284846177190343870827724800000

Offset

0,2

Comments

The numerators are always 1.

Let f(z) = Sum_{n>=0} (1/a(n))*z^(3*n) and g(z) = Sum_{n>=0}(1/b(n))*z^(3*n+1) 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 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

Define W(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^(3*n+1)/(a(n)*(3*n+1)). Then W(z) satisfies the o.d.e. W'''(z) + z*W'(z) = 0 with W(0) = 1, W'(0) = -1, and W''(0) = 0. The function 1/W(z) is the e.g.f. of A117226, which is the number of permutations of [n] avoiding the consecutive pattern 1243. In other words, Sum_{n >= 0} A117226(n)*z^n/n! = 1/W(z). See Theorem 4.3 (Case 1243 with u = 0) in Elizalde and Noy (2003). - Petros Hadjicostas, Nov 01 2019

If y = Sum_{n >= 0} a(n)*x^(3*n+1)/(3*n+1)!, then y' = 1 + x^2*y. - Michael Somos, May 22 2022

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].

Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125; see p. 120.

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

From Peter Bala, Dec 13 2021: (Start)

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

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

a(n) = (3*n + 1)!/(n!*3^n)*hypergeom([-n, 1/3], [4/3], 1).

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

(End)

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

a(n) = A014402(2*n). - Michael Somos, May 22 2022

Example

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

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 13 2021

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. A014402, A117226, A176731.

Column k=3 of A329070.

Sequence in context: A135395 A337756 A141121 * A362174 A225776 A051357

Adjacent sequences: A176727 A176728 A176729 * A176731 A176732 A176733

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Jul 14 2010

Status

approved