A202623 -id:A202623 - cp's OEIS frontend

A073005 Decimal expansion of Gamma(1/3).

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

Nesterenko proves that this constant is transcendental (he cites Chudnovsky as the first to show this); in fact it is algebraically independent of Pi and exp(sqrt(3)*Pi) over Q. - Charles R Greathouse IV, Nov 11 2013

			Gamma(1/3) = 2.6789385347077476336556929409746776441286893779573011009...

H. B. Dwight, Tables of Integrals and other Mathematical Data. 860.18, 860.19 in Definite Integrals. New York, U.S.A.: Macmillan Publishing, 1961, p. 230.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:8 at page 413.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 8.
Yu. V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics 187:9 (1996), pp. 1319-1348. (English translation)
Andrea Pinos, Gamma of reciprocal by Laplace.
Simon Plouffe, GAMMA(1/3).
Wikipedia, Particular values of the Gamma function: General rational arguments.
Index entries for transcendental numbers.
Index to sequences related to the Gamma function.

Cf. A073006, A202623, A203145, A256165.

R:= RealField(100); SetDefaultRealField(R); Gamma(1/3); // G. C. Greubel, Mar 10 2018

Mathematica
```
RealDigits[ N[ Gamma[1/3], 110]][[1]]
```

default(realprecision, 1080); x=gamma(1/3); for (n=1, 1000, d=floor(x); x=(x-d)*10; write("b073005.txt", n, " ", d)); \\ Harry J. Smith, Apr 19 2009

this * A073006 = A186706. - R. J. Mathar, Jan 15 2021
From Amiram Eldar, Jun 25 2021: (Start)
Equals 2^(7/9) * Pi^(1/3) * K((sqrt(3)-1)/(2*sqrt(2)))^(1/3)/3^(1/12), where K is the complete elliptic integral of the first kind.
Equals 2^(7/9) * Pi^(2/3) /(AGM(2, sqrt(2+sqrt(3)))^(1/3) * 3^(1/12)), where AGM is the arithmetic-geometric mean. (End)
From Andrea Pinos, Aug 12 2023: (Start)
Equals Integral_{x=0..oo} 3*exp(-(x^3)) dx = 3*A202623.
General result: Gamma(1/n) = Integral_{x=0..oo} n*exp(-(x^n)) dx. (End)
Equals 3*A202623 = exp(A256165). - Hugo Pfoertner, Jun 28 2024
Equals (2^(1/3)*Pi*C*3^(1/2))^(1/3), where C = A118292 = Integral {0..1} 2/sqrt(1-x^3) is the transcendental butterfly constant. - Jan Lügering, Feb 08 2025

A068467 Decimal expansion of (1/4)! = Gamma(5/4).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Mar 10 2002

			0.906402477055477077982671288966918000748791920720...

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. M. Borwein and I. J. Zucker, Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind, IMA Journal of Numerical Analysis, vol. 12, no. 4, pp. 519-526, 1992.
Greg Martin, A product of Gamma function values at fractions with the same denominator, arXiv:0907.4384v1 [math.CA], 24-July-2009
Albert Nijenhuis, Small Gamma Products with Simple Values, arXiv:0907.1689v1 [math.CA], 9-July-2009.
Raimundas Vidunas, Expressions for values of the gamma function, arXiv:math/0403510 [math.CA], 30-March-2004.
Wikipedia, Particular values of the Gamma function
Index to sequences related to the Gamma function

Cf. A202623.

SetDefaultRealField(RealField(100)); Gamma(5/4); // G. C. Greubel, Mar 11 2018

evalf(GAMMA(5/4)) ; # R. J. Mathar, Jan 10 2013

RealDigits[Gamma[5/4],10,120][[1]] (* Harvey P. Dale, Aug 23 2013 *)

PARI
```
gamma(5/4) \\ Altug Alkan, Sep 18 2016
```

2^(3/4)*(2/e^(16*Pi) + 1)* Pi^(3/4)/(2^(13/16)/(sqrt(2) - 1)^(1/4) + 2^(1/4) + 1) is a very good approximation (~88 digits) which becomes exact if you replace (2/e^(16*Pi) + 1) by EllipticTheta[3,0,exp(-(16*Pi))]. [R. W. Gosper, Posting to Math Fun Mailing List, Dec 27 2011.]
Equals A068466 /4 . - R. J. Mathar, Jan 10 2013
Also equals integral_{0..oo} exp(-x^4) dx. - Jean-François Alcover, Mar 29 2013
Equals 2^(-5/4)*Pi^(3/4)*Product_{k>=1} tanh(Pi*k/2). - Keshav Raghavan, Aug 25 2016

Removed leading zero and adjusted offset, R. J. Mathar, Feb 06 2009
Additional reference from Joerg Arndt, Dec 28 2011
Edited by N. J. A. Sloane, Dec 28 2011

A203125 Decimal expansion of (1/8)! = Gamma(9/8).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 29 2011

			.94174269984970148808740373015189170307630244851863449262289...

Index to sequences related to the Gamma function

Cf. A068467, A202623, A203126, A203142.

RealDigits[(1/8)!,10,120][[1]] (* Harvey P. Dale, May 30 2014 *)

gamma(9/8) \\ Charles R Greathouse IV, Mar 25 2014

Equals A203142/8. - R. J. Mathar, Jan 15 2021
A203144 *this *A231863 *A011006 = A068467. - R. J. Mathar, Jan 15 2021
Equals Integral_{x=0..oo} exp(-x^8) dx. - Ilya Gutkovskiy, Sep 18 2021

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

A376859 Decimal expansion of Product_{k=1..4} Gamma(k/3).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 09 2024

			3.23937134071697320618006601163079489801213782...

Eric Weisstein's World of Mathematics, Gamma Function.
Index to sequences related to the Gamma function.

Cf. A002194, A019692, A202623.

Other identities for Product_{k=1..m} Gamma(k/3): A073005 (m = 1), A186706 (m = 2 and m = 3), A376911 (m = 5 and m = 6), A376912 (m = 7), A376913 (m = 8).

First[RealDigits[2*Pi*Gamma[4/3]/Sqrt[3], 10, 100]]

Equals 2*Pi*Gamma(1/3)/(3*sqrt(3)) = 2*Pi*Gamma(4/3)/sqrt(3) = A186706*A202623 (cf. eq. 86 in Weisstein link).

A256923 Decimal expansion of Sum_{k>=1} (zeta(2k)/k)*(1/3)^(2k).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 13 2015

			0.189958633407180946467791617427446722751559110541442648...

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights (2011) p. 272.

Eric Weisstein's MathWorld, Riemann Zeta Function
Wikipedia, Riemann Zeta Function

Cf. A073006, A202623, A248897, A256924.

RealDigits[Log[2*Pi/(3*Sqrt[3])], 10, 105] // First

Equals log(Gamma(2/3)*Gamma(4/3)).
Equals log(2*Pi/(3*sqrt(3))).
Equals log(A248897).
Equals -Sum_{k>=1} log(1 - 1/(3*k)^2). - Amiram Eldar, Aug 12 2020

A335828 Numerators of coefficients in a power series expansion of the distance between two bodies falling freely towards each other along a straight line under the influence of their mutual gravitational attraction.

Original entry on oeis.org

1, 1, 11, 73, 887, 136883, 7680089, 26838347, 14893630313, 1908777537383, 2422889987331397, 233104477447558811, 2430782624763507659, 14420190617640617313953, 4515429325405165295004389, 812454316441781379614873497, 166481868581561511154267399013

Offset: 1

Amiram Eldar, Jun 25 2020

Consider two point objects with masses m_1 and m_2 that are starting to fall towards each other from rest at time t = 0 and initial distance r_0. Foong (2008) gave the solution for the distance as a function of time, r(t) = r_0 * f(t/t_0), where t_0 = sqrt(r_0^3/(G*(m1+m2))), G is the gravitational constant (A070058), and f(x) = 1 - Sum_{n>=1} c(n) * x^(2*n) is a dimensionless function. c(n) are the rational coefficients whose numerators are given in this sequence. The denominators are given in A335829. The collision occurs when f(x) = 0, at x = Pi/(2*sqrt(2)) (A093954), which corresponds to the time t = (Pi/(2*sqrt(2))) * t_0.

A similar expansion was given by Ernst Meissel in his study of the three-body problem in 1882. In Meissel's expansion the coefficients are c(n)/2^n.

			The series begins with f(x) = 1 - (1/2)*x^2 - (1/12)*x^4 - (11/360)*x^6 - ...

Sudhir Ranjan Jain, Mechanics, Waves and Thermodynamics: An Example-based Approach, Cambridge University Press, 2016. See page 97.
Ernst Meissel, Über Reihen, denen man bei der numerischen Lösung des Problems der Dreikörperproblems begegnet, wenn die Anfangsgeschwindigkeiten Null sind, in: Jahresbericht über die Realschule in Kiel: Während des Schuljahres 1881/82, A. F. Jensen, Kiel, 1882, pp. 1-11.

Amiram Eldar, Table of n, a(n) for n = 1..243
S. K. Foong, From Moon-fall to motions under inverse square laws, European journal of physics, Vol. 29, No. 5 (2008), pp. 987-1003, alternative link.
Jaak Peetre, Ernst Meissel and the Pythagorean problem - the Drei-Körper-Problem in the Nachlass Meissel, draft, 1997.

Cf. A070058, A093954, A202623, A335829 (denominators).

c[1] = 1/2; c[n_] := c[n] = (2*Sum[(n - k)*(2*n - 2*k - 1)*c[n - k]*c[k], {k, 1, n - 1}] - Sum[(n - m)*(2*n - 2*m - 1)*c[n - m]*c[m - k]*c[k], {m, 2, n - 1}, {k, 1, m - 1}])/(n*(2*n - 1)); Numerator @ Array[c, 17]
(* or *)
Quiet[-Numerator @ CoefficientList[AsymptoticDSolveValue[{y[x]*y'[x]^2 == 2*(1-y[x]), y[0] == 1}, y[x], {x, 0, 25}], x][[3;;-1;;2]]] (* requires Mathematica 11.3+ *)

a(n) = numerator(c(n)), c(1) = 1/2, c(n) = (2 * Sum_{k=1..n-1} (n-k)*(2*n-2*k-1)*c(n-k)*c(k) - Sum_{m=2..n-1} (n-m)*(2*n-2*m-1)*c(n-m) * Sum_{k=1..m-1} c(m-k)*c(k))/(n*(2*n - 1)).

c(n) ~ c_0 * n^(-5/3) * (Pi/(2*sqrt(2)))^(-2*n), where c_0 = (3*Pi)^(2/3) / (18*Gamma(4/3)) = 0.277587...

A371856 Decimal expansion of Integral_{x=0..oo} exp(-x^5) dx.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 09 2024

			0.91816874239976061064095165518583040068682...

Index to sequences related to the Gamma function

Decimal expansion of Integral_{x=0..oo} exp(-x^k) dx: A019704 (k=2), A202623 (k=3), A068467 (k=4), this sequence (k=5), A203126 (k=6), A371857 (k=7), A203125 (k=8).

Cf. A175380.

Mathematica
```
RealDigits[Gamma[6/5], 10, 104][[1]]
```

Equals Gamma(6/5).

Equals A175380 / 5.

A371857 Decimal expansion of Integral_{x=0..oo} exp(-x^7) dx.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 09 2024

			0.9354375628925463482448704784898566089...

Index to sequences related to the Gamma function

Decimal expansion of Integral_{x=0..oo} exp(-x^k) dx: A019704 (k=2), A202623 (k=3), A068467 (k=4), A371856 (k=5), A203126 (k=6), this sequence (k=7), A203125 (k=8).

Cf. A220086.

Mathematica
```
RealDigits[Gamma[8/7], 10, 106][[1]]
```

Equals Gamma(8/7).

Equals A220086 / 7.

cp's OEIS Frontend

A073005 Decimal expansion of Gamma(1/3).

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A068467 Decimal expansion of (1/4)! = Gamma(5/4).

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A203125 Decimal expansion of (1/8)! = Gamma(9/8).

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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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A376859 Decimal expansion of Product_{k=1..4} Gamma(k/3).

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A256923 Decimal expansion of Sum_{k>=1} (zeta(2k)/k)*(1/3)^(2k).

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