A073006 -id:A073006 - cp's OEIS frontend

A034164 Related to triple factorial numbers 2*A034000(n+1).

1, 5, 30, 198, 1386, 10098, 75735, 580635, 4528953, 35819901, 286559208, 2314516680, 18846778680, 154543585176, 1274984577702, 10574872085646, 88123934047050, 737458184920050, 6194648753328420, 52212039492339540, 441429061162507020, 3742550735942994300

Offset: 0

Wolfdieter Lang

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A004990, A025748, A034000, A073006, A185047.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1 -3*x -(1-9*x)^(1/3))/(3*x)^2 )); // G. C. Greubel, Sep 17 2019

seq(coeff(series((1-3*x-(1-9*x)^(1/3))/(3*x)^2, x, n+2), x, n), n = 0..32); # G. C. Greubel, Sep 17 2019

CoefficientList[Series[ HypergeometricPFQ[{1, 5/3}, {3}, 9 x], {x, 0, 20}], x]
Table[FullSimplify[3^(2*n+1) * Gamma[n+5/3] / ((n+2) * Gamma[2/3] * Gamma[n+2])],{n,0,20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1 -3x -(1-9 x)^(1/3))/(3 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 10 2014 *)

my(x='x+O('x^30)); Vec((1 -3*x -(1-9*x)^(1/3))/(3*x)^2) \\ G. C. Greubel, Sep 17 2019

def A034164_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1 -3*x -(1-9*x)^(1/3))/(3*x)^2).list()
A034164_list(30) # G. C. Greubel, Sep 17 2019

a(n) = 3^n*(3*n+2)!!!/(n+2)!, where (3*n+2)!!! = 2*A034000(n+1).
G.f.: (1 - 3*x - (1-9*x)^(1/3))/(3*x)^2.
G.f.: 2F1( (1, 5/3); 3; 9 x ). - Olivier Gérard, Feb 15 2011
D-finite with recurrence: (n+2)*a(n) - 3*(3*n+2)*a(n-1) = 0. - R. J. Mathar, Oct 29 2012
a(n) = 3^(2*n+1) * Gamma(n+5/3) / ((n+2) * Gamma(2/3) * Gamma(n+2)). - Vaclav Kotesovec, Feb 09 2014
Integral representation as the n-th moment of a positive function on (0,9): a(n) = Integral_{x=0..9} x^n*W(x) dx, n >= 0, where W(x) = (1/18)*9^(1/3)*sqrt(3)*x^(2/3)*(1-x/9)^(1/3)/Pi. This representation is unique as W(x) is the solution of the Hausdorff moment problem. - Karol A. Penson, Nov 07 2015
Sum_{n>=0} 1/a(n) = 15/16 + (27/64)*(Pi*sqrt(3)/3 - log(3)). - Amiram Eldar, Dec 02 2022
a(n) ~ 3^(2*n+1) * n^(-4/3) / Gamma(2/3). - Amiram Eldar, Aug 19 2025

A284867 Decimal expansion of Ai(0), where Ai is the Airy function of the first kind.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 04 2017

			0.35502805388781723926006318600418317639797917419917724058332651030081...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 56, page 559.

Simon Plouffe, Airy Functions Constant C1.
Eric Weisstein's MathWorld, Airy Functions.
Wikipedia, Airy Function.

Cf. A096714, A096715, A269892, A269893, A073006 (Gamma(2/3)), A284868 (Ai'(0)).

Mathematica
```
RealDigits[AiryAi[0], 10, 105][[1]]
```

airy(0)[1] \\ Charles R Greathouse IV, Apr 26 2019

Ai(0) = 1/(3^(2/3)*Gamma(2/3)).

A030652 Continued fraction for Gamma(2/3).

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 8, 1, 1, 4, 4, 1, 6, 12, 1, 5, 1, 1, 1, 3, 2, 1, 1, 1, 18, 9, 1, 42, 1, 1, 2, 1, 1, 10, 3, 2, 4, 6, 2, 11, 1, 1, 8, 65, 9, 4, 1, 11, 2, 3, 1, 4, 3, 1, 2, 1, 2, 1, 5, 1, 1, 1, 2, 1, 14, 1, 5, 1, 6, 2, 7, 1, 29, 1, 1, 1, 3, 1, 2, 1, 26, 1, 1, 7, 13, 1, 2, 2, 8, 3, 4, 2, 2, 2

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it)

			1.354117939426400416945288028... = 1 + 1/(2 + 1/(1 + 1/(4 + 1/(1 + ...)))). - _Harry J. Smith_, May 14 2009

Harry J. Smith, Table of n, a(n) for n = 0..4999
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A030651, A073006 (decimal expansion).

ContinuedFraction[Gamma[2/3],100] (* Harvey P. Dale, Sep 05 2021 *)

{ allocatemem(932245000); default(realprecision, 5200); x=contfrac(gamma(2/3)); for (n=1, 5000, write("b030652.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 14 2009

Note that 3*GAMMA(1/3)*GAMMA(2/3) = 2*Pi*sqrt(3).

Offset changed by Andrew Howroyd, Aug 03 2024

A204067 Decimal expansion of the Fresnel Integral, Integral_{x >= 0} cos(x^3) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

			0.7733429420779898501961016...

Muniru A Asiru, Table of n, a(n) for n = 0..2000
R. J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 [math.CA], 2012, eq. (3.8).
Wikipedia, Fresnel Integral.

Cf. A010469, A019670, A073005, A073006, A204068, A217481.

evalf(int(cos(x^3),x=0..infinity),120); # Muniru A Asiru, Sep 26 2018

RealDigits[Gamma[1/3]/(2*Sqrt[3]), 10, 120][[1]] (* Amiram Eldar, May 26 2023 *)

Pi/(3*gamma(2/3)) \\ Gheorghe Coserea, Sep 26 2018

intnum(x=[0, -2/3], [oo, I], cos(x)/x^(2/3))/3 \\ Gheorghe Coserea, Sep 26 2018

Equals Pi/(3*Gamma(2/3)) = A019670 / A073006.

Equals Gamma(1/3)/(2*sqrt(3)) = A073005 / A010469. - Amiram Eldar, May 26 2023

A290570 Decimal expansion of Integral_{0..Pi/2} dtheta/(cos(theta)^3 + sin(theta)^3)^(2/3).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Aug 07 2017

			1.766638750285449957313689499648438702571868538202557530126905241835453...

Oscar S. Adams, Elliptic Functions Applied to Conformal World Maps, Special Publication No. 112 of the U.S. Coast and Geodetic Survey, 1925. See constant K p. 9 and previous pages.

StackExchange, Question 2380131 Evaluate in closed form Integral_{0..Pi/2} dtheta/(cos(theta)^3+sin(theta)^3)^(2/3), August 2017.

Cf. A073005 (Gamma(1/3)), A073006 (Gamma(2/3)), A197374 (Beta(1/3,1/3)).

Cf. A104133, A104134, A197374.

RealDigits[(1/3)*Gamma[1/3]^2/Gamma[2/3], 10, 105]

(1/3)*gamma(1/3)^2/gamma(2/3) \\ Michel Marcus, Aug 07 2017

Equals (1/3)*Beta(1/3,1/3).
Equals (1/3)*Gamma(1/3)^2/Gamma(2/3).
Equals A197374/3. - Michel Marcus, Jun 08 2020
From Peter Bala, Mar 01 2022: (Start)
Equals 2*Sum_{n >= 0} (1/(3*n+1) + 1/(3*n-2))*binomial(1/3,n). Cf. A002580 and A175576.
Equals Sum_{n >= 0} (-1)^n*(1/(3*n+1) - 1/(3*n-2))*binomial(1/3,n).
Equals hypergeom([1/3, 2/3], [4/3], 1) = (3/2)*hypergeom([-1/3, -2/3], [4/3], 1) = 2*hypergeom([1/3, 2/3], [4/3], -1) = hypergeom([-1/3, -2/3, 5/6], [4/3, -1/6], -1). (End)

A203126 Decimal expansion of (1/6)! = Gamma(7/6).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 29 2011

			.92771933363003920070834948253462101856646651914547557693612...

Index to sequences related to the Gamma function

Cf. A068467, A175379, A202623, A203125.

RealDigits[(1/6)!,10,120][[1]] (* Harvey P. Dale, Aug 10 2013 *)

gamma(7/6) \\ Charles R Greathouse IV, Mar 25 2014

Equals A175379/6. - R. J. Mathar, Jan 15 2021
A073006 * this * A231863 * A329219 = A202623. - R. J. Mathar, Jan 15 2021
Equals Integral_{x=0..oo} exp(-x^6) dx. - Ilya Gutkovskiy, Sep 18 2021

A204068 Decimal expansion of the Fresnel Integral Integral_{x>=0} sin(x^3) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

Imaginary part associated with A204067.

			0.446489755784624605609282...

R. J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 [math.CA], 2012, eq. (3.8).
Wikipedia, Fresnel Integral.

Cf. A073005, A073010, A073006, A202623, A204067

Maple
```
evalf(Pi/GAMMA(2/3)/3^(3/2) ) ;
```

RealDigits[Gamma[1/3]/6, 10, 120][[1]] (* Amiram Eldar, May 26 2023 *)

Equals Pi/(Gamma(2/3)* 3^(3/2)) = A073010 / A073006.
(this value)^2 + A204067^2 = A202623^2.
Equals Gamma(1/3)/6 = A073005 / 6. - Amiram Eldar, May 26 2023

A205885 Decimal expansion of the Fresnel Integral int_{x=0..infinity} x*cos(x^3) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

			0.2256863232377334028242146713590856309...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Fresnel Integral

C := ComplexField(); [Gamma(2/3)/6]; // G. C. Greubel, Dec 13 2017

Maple

evalf(GAMMA(2/3)/6) ;

Mathematica

RealDigits[Gamma[2/3]/6, 10, 100][[1]] (* G. C. Greubel, Dec 13 2017 *)

PARI

gamma(2/3)/6 \\ G. C. Greubel, Dec 13 2017

Formula

Equals A073006/6.

cp's OEIS Frontend

A034164 Related to triple factorial numbers 2*A034000(n+1).

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A284867 Decimal expansion of Ai(0), where Ai is the Airy function of the first kind.

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A030652 Continued fraction for Gamma(2/3).

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A204067 Decimal expansion of the Fresnel Integral, Integral_{x >= 0} cos(x^3) dx.

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A290570 Decimal expansion of Integral_{0..Pi/2} dtheta/(cos(theta)^3 + sin(theta)^3)^(2/3).

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A203126 Decimal expansion of (1/6)! = Gamma(7/6).

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A204068 Decimal expansion of the Fresnel Integral Integral_{x>=0} sin(x^3) dx.

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