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

Offset

0,3

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

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.

Formula

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

Example

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) - ...

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 *)

Prog

(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 */
(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
(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

Crossrefs

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

Sequence in context: A336723 A334805 A324980 * A181493 A360524 A051784

Adjacent sequences: A014399 A014400 A014401 * A014403 A014404 A014405

Keyword

nonn

Author

N. J. A. Sloane

Status

approved