A014403 -id:A014403 - 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

A060507 Denominators of the asymptotic expansion of the Airy function Ai(x).

Original entry on oeis.org

1, 72, 3456, 746496, 214990848, 1719926784, 743008370688, 53496602689536, 10271347716390912, 6655833320221310976, 958439998111868780544, 23002559954684850733056

Offset: 0

Michael Praehofer (praehofer(AT)ma.tum.de), Mar 22 2001

The series arises in the asymptotic expansion of the Airy function A(x) for large |x| as Ai(x)~pi^(-1/2)/2*x^(-1/4)*exp(-z)*sum((-1)^k*c(k)*z^(-k),k=0..infinity), where z=2/3*x^(3/2). a(k) is the denominator of the fully canceled c(k).

			a(2) = 3456 because for k=2, product((2*l+1),l=k..3*k-1)/216^k/k! =  385/3456 and we take the denominator of the fully canceled fraction.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
NIST's Digital Library of Mathematical Functions, Airy and Related Functions (Poincaré-Type Expansions) by Frank W. J. Olver.

Cf. A060506, A014402, A014403.

a[ n_] := If[ n<0, 0, Denominator[ Product[k, {k, 1, 6 n - 1, 2}] / n! / 216^n]] (* Michael Somos, Oct 14 2011 *)

a(k)=denom(product((2*l+1), l=k..3*k-1)/216^k/k!).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula