A014402 -id:A014402 - cp's OEIS frontend

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

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

A203433 Vandermonde determinant of the first n terms of (2,3,5,6,8,9,...) = (j+floor((j+1)/2)).

Original entry on oeis.org

1, 1, 6, 72, 12960, 6531840, 84652646400, 3839844040704000, 6568897997313146880000, 46482573252667397426380800000, 16698920220108665726304214056960000000, 28359415513133792655802758561911537664000000000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

Each term divides its successor, as in A014402, and each term is divisible by the corresponding superfactorial, A000178(n), as in A203434.

G. C. Greubel, Table of n, a(n) for n = 1..41

Cf. A000178, A007494, A014402, A203434.

A203433:= func< n | n eq 1 select 1 else (&*[(&*[k-j+Floor(k/2)-Floor(j/2): j in [0..k-1]]) : k in [1..n-1]]) >;
[A203433(n): n in [1..25]]; // G. C. Greubel, Sep 20 2023

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}]             (* this sequence *)
Table[v[n+1]/v[n], {n,z}]      (* A014402 *)
Table[v[n]/d[n], {n,z}]        (* A203434 *)

def A203433(n): return product(product(k-j+(k//2)-(j//2) for j in range(k)) for k in range(1,n))
[A203433(n) for n in range(1,31)] # G. C. Greubel, Sep 20 2023

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!).

A203434 a(n) = A203433(n)/A000178(n) where A000178=(superfactorials).

Original entry on oeis.org

1, 1, 3, 6, 45, 189, 3402, 30618, 1299078, 25332021, 2507870079, 106698472452, 24487299427734, 2283997201168644, 1209640056157393380, 248218139523497121576, 302358334494179897593596, 136861610819571430116630660

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..100
R. Chapman, A polynomial taking integer values, Mathematics Magazine, 29 (1996), 121.

Cf. A000178, A007494, A014402, A203432, A203433.

Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;
f:= func< k | (&*[k+1-j+Floor((k+2)/2)-Floor((j+1)/2): j in [1..k]]) >;
[1] cat [(&*[f(k): k in [1..n-1]])/Barnes(n): n in [2..20]]; // G. C. Greubel, Sep 19 2023

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}]      (* A014402 *)
Table[v[n]/d[n], {n,z}]        (* A203434 *)

def barnes(n): return product(factorial(j) for j in range(n))
def f(k): return product(k-j+(k//2)-(j//2) for j in range(k))
[product(f(k) for k in range(1, n) )//barnes(n) for n in range(1,31)] # G. C. Greubel, Sep 19 2023

A014403 Numbers found in denominators of expansion of Airy function Bi(x).

Original entry on oeis.org

1, 1, 6, 4, 180, 168, 12960, 15120, 1710720, 2358720, 359251200, 566092800, 109930867200, 193603737600, 46170964224000, 89444926771200, 25486372251648000, 53666956062720000

Offset: 0

N. J. A. Sloane

nonn

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

			Mathematica gives the series as 1/(3^(1/6)*Gamma(2/3)) + 3^(1/6)*x/Gamma(1/3) + x^3/(6*3^(1/6)*Gamma(2/3)) + x^4/(4*3^(5/6)*Gamma(1/3)) + x^6/(180*3^(1/6)*Gamma(2/3)) + x^7/(168*3^(5/6)*Gamma(1/3)) + x^9/(12960*3^(1/6)*Gamma(2/3)) + ...

NIST's Digital Library of Mathematical Functions, Airy and Related Functions (Maclaurin Series) by Frank W. J. Olver.

Cf. A014402.

Mathematica
```
Series[ AiryBi[ x ], {x, 0, 30} ]
```

A385253 Decimal expansion of the function AiryAi(x) at x=1.

Original entry on oeis.org

1, 3, 5, 2, 9, 2, 4, 1, 6, 3, 1, 2, 8, 8, 1, 4, 1, 5, 5, 2, 4, 1, 4, 7, 4, 2, 3, 5, 1, 5, 4, 6, 6, 3, 0, 6, 1, 7, 4, 9, 4, 4, 1, 4, 2, 9, 8, 8, 3, 3, 0, 7, 0, 6, 0, 0, 9, 1, 0, 2, 0, 5, 4, 7, 5, 7, 6, 3, 3, 5, 3, 4, 8, 0, 2, 2, 6, 5, 7, 2, 3, 6, 6, 3, 4, 8, 7, 1, 0, 9, 9, 0, 8, 7, 4, 8, 6, 8, 3, 2, 1, 3, 0, 5, 3

Offset: 0

Artur Jasinski, Jul 29 2025

Second derivative at x=1 has that same value.

			0.1352924163128814155241474235154663...

Cf. A014402, A176730, A284867, A284868, A269893.

AiryAi(1) ; evalf(%) ; # R. J. Mathar, Jul 31 2025

Mathematica
```
RealDigits[AiryAi[1], 10, 105][[1]]
```

airy(1)[1] \\ Michel Marcus, Jul 29 2025

Equals AiryAi''(1).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A203433 Vandermonde determinant of the first n terms of (2,3,5,6,8,9,...) = (j+floor((j+1)/2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A203434 a(n) = A203433(n)/A000178(n) where A000178=(superfactorials).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

A014403 Numbers found in denominators of expansion of Airy function Bi(x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A385253 Decimal expansion of the function AiryAi(x) at x=1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula