A175380 -id:A175380 - cp's OEIS frontend

A269559 Decimal expansion of Psi(log(2)), negated.

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

Offset: 1

Iaroslav V. Blagouchine, Feb 29 2016

Psi(x) is the digamma function (logarithmic derivative of the Gamma function).

			-1.2395972796176185082441275516860842454332895226874208...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A269557, A269558, A269545, A269546, A269547, A257955, A257957, A257958, A257959, A002161.

Cf. A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

MATLAB
```
format long; psi(log(2))
```
Maple
```
evalf(Psi(ln(2)), 120);
```

RealDigits[PolyGamma[Log[2]], 10, 120][[1]]

default(realprecision, 120); psi(log(2))

A340722 Decimal expansion of Gamma(4/5).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 17 2021

			1.164229713725303373636...

DLMF, Gamma function properties, DLMF section 5.5.
Index to sequences related to gamma function.

Cf. A019845, A175380, A340721, A340723.

Maple
```
evalf(GAMMA(4/5),120) ;
```

RealDigits[Gamma[4/5], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

this * A175380 = Pi/A019845. [DLMF (5.5.3)]

this * A340723 * 2^(1/10)/sqrt(2*Pi) = A340721. [DLMF (5.5.5)]

A246745 Decimal expansion of Gamma(2/5).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 02 2014

			2.21815954375768822305905402190767945077056650177146958224...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 4.
Index to sequences related to gamma function

Cf. A175379, A175380 (both constants are mentioned in Finch's Addenda), A340721.

RealDigits[Gamma[2/5], 10, 103] // First

gamma(2/5) \\ Michel Marcus, Sep 02 2014

A340721 Decimal expansion of Gamma(3/5).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 17 2021

			1.489192248812817102..

DLMF, Gamma function properties, DLMF section 5.5.
Eric Weisstein's World of Mathematics, Gamma Function.
Wikipedia, Particular values of the Gamma function.
Index to sequences related to gamma function.

Cf. A019881, A175380, A246745, A256191.

Maple
```
evalf(GAMMA(3/5),120) ;
```

RealDigits[Gamma[3/5], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

this * A246745 = Pi/A019881. [DLMF (5.5.3)]

this * A256191 *2^(7/10)/sqrt(2*Pi) = 2*A175380 [DLMF (5.5.5)]

A340724 Decimal expansion of Gamma(7/10).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 17 2021

			1.29805533264755778568...

DLMF, Gamma function properties, DLMF section 5.5.
Index to sequences related to gamma function

Cf. A246745, A175380, A246745.

Maple
```
evalf(GAMMA(7/10),120) ;
```

RealDigits[Gamma[7/10],10,120][[1]] (* Harvey P. Dale, Mar 02 2023 *)

gamma(.7) \\ Charles R Greathouse IV, Nov 26 2024

this * A340723 = Pi/A019863 [DLMF (5.5.3)]

this * A175380 * 2^(9/10)/sqrt(2*Pi) = 2*A246745. [DLMF (5.5.5)]

A371881 Decimal expansion of Gamma(1/20).

Original entry on oeis.org

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

Offset: 2

Vaclav Kotesovec, Apr 15 2024

			19.4700853112555128640473209677271275456304195833419756810827837553645...

Index to sequences related to gamma function

Cf. A073005, A175380, A256191, A203140, A371983.

evalf(GAMMA(1/20), 130);  # Alois P. Heinz, Apr 15 2024

RealDigits[Gamma[1/20], 10, 120][[1]]
RealDigits[2^(33/40) * 5^(5/16) * (1 + Sqrt[5])^(1/8) * Sqrt[5^(1/4) + Sqrt[2 + Sqrt[5]]] * Sqrt[Pi * Gamma[1/10]] * QPochhammer[E^(-2*Sqrt[5]*Pi)] / E^(Sqrt[5]*Pi/12), 10, 120][[1]]

Equals 2^(33/40) * 5^(5/16) * (1 + sqrt(5))^(1/8) * sqrt(5^(1/4) + sqrt(2 + sqrt(5))) * sqrt(Pi*Gamma(1/10)) * QPochhammer(exp(-2*sqrt(5)*Pi)) / exp(sqrt(5)*Pi/12).

A091545 First column sequence of the array (7,2)-Stirling2 A091747.

Original entry on oeis.org

1, 42, 5544, 1507968, 696681216, 489070213632, 485157651922944, 646229992361361408, 1112808046846264344576, 2405890997281623512973312, 6380422924790865556405223424, 20366309975932442856045473169408, 77025384328976498881563979526701056, 340606249502734078054275917467072069632

Offset: 1

Wolfdieter Lang, Feb 13 2004

Also sixth column (m=5) sequence of triangle A091543.

Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.

Cf. A008548, A034323, A091543, A091747, A175380, A246745.

a[n_] := 5^(2*n) * Pochhammer[1/5, n] * Pochhammer[2/5, n] / 2; Array[a, 15] (* Amiram Eldar, Sep 01 2025 *)

a(n) = Product_{j=0..n-1} ((5*j+2)*(5*j+1))/2, n>=1. From eq.12 of the Blasiak et al. reference with r=7, s=2, k=1.
a(n) = (5^(2*n))*risefac(1/5, n)*risefac(2/5, n)/2, n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac5(5*n-3)*fac5(5*n-4)/2, n>=1, with fac5(5*n-4)/2 = A034323(n) and fac5(5*n-3) = A008548(n) (5-factorials).
E.g.f.: (hypergeom([1/5, 2/5], [], 25*x)-1)/2.
a(n) = A091747(n, 2), n>=1.
D-finite with recurrence a(n) - (5*n-3)*(5*n-4)*a(n-1) = 0. - R. J. Mathar, Jul 27 2022
a(n) ~ Pi * (5/e)^(2*n) * n^(2*n-2/5) / (Gamma(1/5) * Gamma(2/5)). - Amiram Eldar, Sep 01 2025
a(n) ~ sqrt(Pi*(1 + sqrt(5))) * 5^(2*n + 1/4) * n^(2*n - 2/5) / (Gamma(1/10) * 2^(7/10) * exp(2*n)). - Vaclav Kotesovec, Sep 01 2025

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.

A371983 Decimal expansion of Gamma(1/30).

Original entry on oeis.org

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

Offset: 2

Vaclav Kotesovec, Apr 15 2024

			29.4547797456996940196962082886383457347018736055729711046565415567498...

Albert Nijenhuis, Small Gamma Products with Simple Values, arXiv:0907.1689v1 [math.CA], 2009.
R. Vidunas, Expressions for values of the Gamma function, arxiv:math/0403510 [math.CA], 2004.
Index to sequences related to gamma function

Cf. A073005, A175380, A256191, A203140, A371881.

evalf(GAMMA(1/30), 130);  # Alois P. Heinz, Apr 15 2024

RealDigits[Gamma[1/30], 10, 120][[1]]
RealDigits[2^(11/60) * 3^(9/20) * 5^(1/3) * Gamma[1/5] * Gamma[1/3] / ((10 + Sqrt[5] - Sqrt[75 + 30*Sqrt[5]])^(1/4) * Sqrt[Pi]), 10, 120][[1]]

Equals 3^(9/20) * sqrt(5 + sqrt(5)) * sqrt(sqrt(15) + sqrt(5 + 2*sqrt(5))) * Gamma(1/3) * Gamma(1/5) / (sqrt(Pi) * 2^(16/15) * 5^(1/6)).
Equals 2^(11/60) * 3^(9/20) * 5^(1/3) * Gamma(1/5) * Gamma(1/3) / ((10 + sqrt(5) - sqrt(75 + 30*sqrt(5)))^(1/4) * sqrt(Pi)).
Equals 8*Pi^2 / (Gamma(17/30) * Gamma(19/30) * Gamma(23/30)).
Equals Gamma(7/30) * Gamma(11/30) * Gamma(13/30) / (2*Pi*A019815).

A377405 Decimal expansion of Pi*csc(Pi/5).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Oct 27 2024

			5.344796660577975567125921862534413199507254626...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 360.
Index entries for transcendental numbers.

Cf. A000796, A001622, A002163, A010476, A019692, A175380, A345019, A340722, A352324.

Mathematica
```
RealDigits[Pi*Csc[Pi/5],10,100][[1]]
```

Equals Gamma(1/5)*Gamma(4/5) = 2*Pi*sqrt(5)*sqrt(2+phi)/5 (see Finch).
Equals Integral_{x=0..oo} log(1 + x^(-5)) dx (see Shamos).
Equals sqrt(2*(1 + 1/sqrt(5)))*Pi.
Equals 10 * Sum_{n>=1} (-1)^(n+1)/A345019(n). - Amiram Eldar, Oct 27 2024
Equals 5*A352324. - Hugo Pfoertner, Oct 28 2024

cp's OEIS Frontend

A269559 Decimal expansion of Psi(log(2)), negated.

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A340722 Decimal expansion of Gamma(4/5).

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A246745 Decimal expansion of Gamma(2/5).

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A340721 Decimal expansion of Gamma(3/5).

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A340724 Decimal expansion of Gamma(7/10).

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A371881 Decimal expansion of Gamma(1/20).

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A091545 First column sequence of the array (7,2)-Stirling2 A091747.

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