A340722 -id:A340722 - cp's OEIS frontend

A175380 Decimal expansion of Gamma(1/5).

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

Offset: 1

R. J. Mathar, Apr 24 2010

			Equals 4.590843711998803053204758275929152...

G. C. Greubel, Table of n, a(n) for n = 1..5000
R. Vidunas, Expressions for values of the Gamma function, arxiv:math/0403510 [math.CA], 2004.
Wikipedia, Particular values of the Gamma function: General rational arguments
Index to sequences related to the gamma function

SetDefaultRealField(RealField(100)); Gamma(1/5); // G. C. Greubel, Mar 10 2018

Maple
```
evalf(GAMMA(1/5));
```

RealDigits[Gamma[1/5],10,120][[1]] (* Harvey P. Dale, May 26 2011 *)

gamma(1/5) \\ G. C. Greubel, Jan 15 2017

Equals Pi * sqrt(2) * sqrt(5 + sqrt(5)) / (sqrt(5) * Gamma(4/5) ) = A063448 * sqrt(5 + A002163) / (A002163 * Gamma(4/5)) where Gamma(4/5) = A340722.

A256191 Decimal expansion of Gamma(1/10).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 19 2015

			9.513507698668731836292487177265402192550578626088377...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index to sequences related to the Gamma function

Cf. A175380, A246745, A340721, A340722.

Cf. A340723, A340724, A340725.

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

Maple
```
evalf(GAMMA(1/10),100);
```
Mathematica
```
RealDigits[Gamma[1/10],10,100][[1]]
```
PARI
```
gamma(1/10)
```

From Vaclav Kotesovec, Apr 10 2024: (Start)
Equals 5^(1/4) * sqrt(1 + sqrt(5)) * Gamma(1/5) * Gamma(2/5) / (2^(7/10) * sqrt(Pi)).
Equals 2^(4/5) * sqrt(Pi) * Gamma(1/5) / Gamma(3/5). (End)

A025750 5th-order Patalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 10, 150, 2625, 49875, 997500, 20662500, 439078125, 9513359375, 209293906250, 4661546093750, 104884787109375, 2380077861328125, 54401779687500000, 1251240932812500000, 28934946571289062500, 672311993862304687500, 15687279856787109375000, 367412607172119140625000

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015), Article 15.3.3; arXiv preprint, arXiv:1410.5880 [math.CO], 2014.

Cf. A034687, A049393, A340722.

CoefficientList[Series[(6-(1-25x)^(1/5))/5,{x,0,20}],x] (* Harvey P. Dale, Dec 06 2012 *)
a[0] = 1; a[n_] := ((-5)^(n - 1)*Sum[5^(n - k)*StirlingS1[n, k], {k, 1, n}])/n!; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 19 2013, after Vladimir Kruchinin *)
a[n_] := 25^(n-1) * Pochhammer[4/5, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)

a(n):=if n=0 then 1 else (sum((-1)^(n-k-1)*binomial(n+k-1,n-1)*sum(2^j*binomial(k,j)*sum(binomial(j,i-j)*binomial(k-j,n-3*(k-j)-i-1)*5^(3*(k-j)+i),i,j,n-k+j-1),j,0,k),k,0,n-1))/(n); /* Vladimir Kruchinin, Dec 10 2011 */

a(n):=if n=0 then 1 else -binomial(1/5,n)*(-25)^n/5; /* Tani Akinari, Sep 17 2015 */

G.f.: (6-(1-25*x)^(1/5))/5.
a(n) = 5^(n-1)*4*A034301(n-1)/n!, n >= 2, where 4*A034301(n-1) = (5*n-6)(!^5) = Product_{j=2..n} (5*j-6). - Wolfdieter Lang
a(n) = Sum_{k=0..n-1} (-1)^(n-k-1)*binomial(n+k-1,n-1) * Sum_{j=0..k} 2^j*binomial(k,j) * Sum_{i=j..n-k+j-1} binomial(j,i-j)*binomial(k-j,n-3*(k-j)-i-1)*5^(3*(k-j)+i)/n, n > 0, a(0) = 1. - Vladimir Kruchinin, Dec 10 2011
a(n) = ((-5)^(n-1)*Sum_{k=1..n} (5)^(n-k)*stirling1(n,k))/n!, n>0, a(0) = 1. - Vladimir Kruchinin, Mar 19 2013
From Karol A. Penson, Feb 05 2025: (Start)
a(n) without the initial 1 (i.e., a(n) for n >= 1) is given by
a(n+1) = 5^(2*n)*gamma(n + 4/5)/(gamma(4/5)*(n + 1)!), n >= 0.
a(n+1) = Integral_{x=0..25} x^n*W(x) dx, n >= 0,
  where W(x) = sin(Pi/5)*5^(2/5)*(1 - x/25)^(1/5)/(5*Pi*x^(1/5)). W(x) is a positive  function on x = (0, 25), is singular at x = 0 with the singularity (x)^(-1/5), and it goes to zero at x = 25. (End)
a(n) ~ 25^(n-1) / (Gamma(4/5) * n^(6/5)). - Amiram Eldar, Aug 20 2025

A340723 Decimal expansion of Gamma(3/10).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 17 2021

			2.991568987687590628312...

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

Cf. A019863, A340721, A340722, A340724.

Maple
```
evalf(GAMMA(3/10),120) ;
```

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

this * A340724 = Pi/A019863 [DLMF (5.5.3)]

this * A340722 * 2^(1/10)/sqrt(2*Pi) = A340721. [DLMF (5.5.5)]

A340725 Decimal expansion of Gamma(9/10).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 17 2021

			1.06862870211931935489..

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

Cf. A019827, A246745, A256191, A340722.

Maple
```
evalf(GAMMA(9/10),120) ;
```

RealDigits[Gamma[9/10], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

this * A256191 = Pi/A019827 . [DLMF (5.5.3)]

A246745 * this *2^(3/10) /sqrt(2*Pi) = A340722 . [DLMF (5.5.5)]

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

A175380 Decimal expansion of Gamma(1/5).

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A256191 Decimal expansion of Gamma(1/10).

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A025750 5th-order Patalan numbers (generalization of Catalan numbers).

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Maxima

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A340723 Decimal expansion of Gamma(3/10).

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A340725 Decimal expansion of Gamma(9/10).

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

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