A176731 -id:A176731 - cp's OEIS frontend

A014402 Numbers found in denominators of expansion of Airy function Ai(x).

1, 1, 6, 12, 180, 504, 12960, 45360, 1710720, 7076160, 359251200, 1698278400, 109930867200, 580811212800, 46170964224000, 268334780313600, 25486372251648000, 161000868188160000, 17891433320656896000, 121716656350248960000, 15565546988971499520000, 113196490405731532800000

Offset: 0

N. J. A. Sloane

nonn

Although the description is technically correct, this sequence is unsatisfactory because there are gaps in the series.

A014402 arises via Vandermonde determinants as in A203433; see the Mathematica section. - Clark Kimberling, Jan 02 2012

			Mathematica gives the series as 1/(3^(2/3)*Gamma(2/3)) - x/(3^(1/3)*Gamma(1/3)) + x^3/(6*3^(2/3)*Gamma(2/3)) - x^4/(12*3^(1/3)*Gamma(1/3)) + x^6/(180*3^(2/3)*Gamma(2/3)) - x^7/(504*3^(1/3)*Gamma(1/3)) + x^9/(12960*3^(2/3)*Gamma(2/3)) - ...

G. C. Greubel, Table of n, a(n) for n = 0..420
NIST's Digital Library of Mathematical Functions, Airy and Related Functions (Maclaurin Series) by Frank W. J. Olver.

Cf. A014403, A060507, A176730, A176731, A203433, A203434.

A014402:= func< n | n eq 0 select 1 else (&*[n-j+Floor(n/2)-Floor(j/2): j in [0..n-1]]) >;
[A014402(n): n in [0..25]]; // G. C. Greubel, Sep 20 2023

Series[ AiryAi[ x ], {x, 0, 30} ]
a[ n_] := If[ n<0, 0, (n + Quotient[ n, 2])! / Product[ 3 k + 1 + Mod[n, 2], {k, 0, Quotient[ n, 2] - 1}]]; (* Michael Somos, Oct 14 2011 *)
(* Next, A014402 generated in via Vandermonde determinants based on A007494 *)
f[j_]:= j + Floor[(j+1)/2]; z = 20;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,z}]             (* A203433 *)
Table[v[n+1]/v[n], {n,z}]      (* this sequence *)
Table[v[n]/d[n], {n,z}]        (* A203434 *)
(* Clark Kimberling, Jan 02 2012 *)

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

def A014402(n): return product(n-j+(n//2)-(j//2) for j in range(n))
[A014402(n) for n in range(31)] # G. C. Greubel, Sep 20 2023

a(2*n) = A176730(n). a(2*n + 1) = A176731(n). - Michael Somos, Oct 14 2011

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

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Jul 14 2010

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

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

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.

Cf. A014402, A117226, A176731.

Column k=3 of A329070.

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

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 *)

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

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

cp's OEIS Frontend

A014402 Numbers found in denominators of expansion of Airy function Ai(x).

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