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

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Examples

			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)) - ...
		

Links

• 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.

Crossrefs

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

Programs

Magma

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

Mathematica

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 *)

PARI

{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 */

SageMath

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

Formula

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