A175379 -id:A175379 - 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))

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

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

A034789 Related to sextic factorial numbers A008542.

Original entry on oeis.org

1, 21, 546, 15561, 466830, 14471730, 458960580, 14801478705, 483514971030, 15955994043990, 530899438190940, 17785131179396490, 599222112044281740, 20287948650642110340, 689790254121831751560, 23539092421907508521985, 805867752326480585870310, 27668126163209166781547310

Offset: 1

Wolfdieter Lang

Convolution of A004993(n-1) with A025751(n), n >= 1.

Michael De Vlieger, Table of n, a(n) for n = 1..645
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Index entries for sequences related to factorial numbers.

Cf. A004993, A008542, A025751, A034687, A175379.

List([1..20], n-> 6^(n-1)*Product([1..n], j-> 6*j-5)/Factorial(n) ); # G. C. Greubel, Nov 11 2019

[6^(n-1)*(&*[6*j-5: j in [1..n]])/Factorial(n): n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( 6^(n-1)*mul(6*j-5, j=1..n)/n!, n=1..20); # G. C. Greubel, Nov 11 2019

Rest@ CoefficientList[Series[(-1 + (1 - 36 x)^(-1/6))/6, {x, 0, 16}], x] (* Michael De Vlieger, Oct 13 2019 *)
Table[6^(2*n-1)*Pochhammer[1/6, n]/n!, {n, 20}] (* G. C. Greubel, Nov 11 2019 *)

vector(20, n, 6^(n-1)*prod(j=1,n, 6*j-5)/n! ) \\ G. C. Greubel, Nov 11 2019

[6^(n-1)*product( (6*j-5) for j in (1..n))/factorial(n) for n in (1..20)] # G. C. Greubel, Nov 11 2019

a(n) = 6^(n-1)*A008542(n)/n!.
G.f.: (-1+(1-36*x)^(-1/6))/6.
D-finite with recurrence: n*a(n) + 6*(-6*n+5)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
a(n) ~ 6^(2*n-1) * n^(-5/6) / Gamma(1/6). - Amiram Eldar, Aug 18 2025

A091546 First column of the array A092077 ((8,2)-Stirling2).

Original entry on oeis.org

1, 56, 10192, 3872960, 2517424000, 2497284608000, 3511182158848000, 6643156644540416000, 16275733779124019200000, 50129260039701979136000000, 189588861470152885092352000000, 863766852858016544480755712000000, 4666068539139005373285042356224000000, 29489553167358513959161467691335680000000

Offset: 1

Wolfdieter Lang, Feb 13 2004

Also seventh column (m=6) 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. A007559, A008542, A034689, A091543, A092077.

Cf. A073005, A175379.

a[n_] := 6^(2*n) * Pochhammer[1/6, n] * Pochhammer[1/3, n] / 2; Array[a, 20] (* Amiram Eldar, Aug 30 2025 *)

a(n) = (2^(n-1))*Product_{j=0..n-1} ((3*j+1)*(6*j+1)), n>=1. From eq.12 of the Blasiak et al. reference with r=8, s=2, k=1.
a(n) = (6^(2*n))*risefac(1/6, n)*risefac(1/3, n)/2, n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac6(6*n-5)*fac6(6*n-4)/2, n>=1, with fac6(6*n-5) = A008542(n) and fac6(6*n-4)/2 = A034689(n)= (2^(n-1))*A007559(n), (6-factorials).
a(n) ~ Pi * (6/e)^(2*n) * n^(2*n-1/2) / (Gamma(1/6) * Gamma(1/3)). - Amiram Eldar, Aug 30 2025

A243308 Decimal expansion of h_3, a constant related to certain evaluations of the gamma function from elliptic integrals.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 03 2014

			1.0174087975959560086695387533500634259952569...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.5.4 Gamma function, p. 34.

Eric Weisstein's MathWorld, Gamma function

Cf. A073005, A175379, A175574.

Re(evalf(4*EllipticK(sqrt((4*sqrt(3)-7)))/(sqrt(2+sqrt(3))*Pi), 120)); # Vaclav Kotesovec, Apr 22 2015

RealDigits[4*EllipticK[4*Sqrt[3]-7]/(Sqrt[2+Sqrt[3]]*Pi), 10, 103] // First
RealDigits[1/ArithmeticGeometricMean[1, Sqrt[2 + Sqrt[3]]/2], 10, 103][[1]] (* Jan Mangaldan, Jan 06 2017 *)
RealDigits[2 EllipticK[(2 - Sqrt[3])/4]/Pi, 10, 103][[1]] (* Jan Mangaldan, Jan 06 2017 *)

4*K(4*sqrt(3)-7)/(sqrt(2+sqrt(3))*Pi), where K is the complete elliptic integral of the first kind.
3^(1/4)*GAMMA(1/3)^3/(2*2^(1/3)*Pi^2), where GAMMA is the Euler Gamma function.
GAMMA(1/6)^(3/2)/(2^(5/6)*sqrt(3)*Pi^(5/4)).

A357318 Decimal expansion of 1/(2*L), where L is the conjectured Landau's constant A081760.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Sep 23 2022

			0.9203713733179424976555185645431729947262...

Markus Faulhuber, An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a "Weltkonstante"-or-how Ramanujan split temperatures, The Ramanujan Journal volume 54, pages 1-27 (2021). See p. 23.
Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 2.

Cf. A004117.
Cf. A081760.
Cf. A073005, A175379, A203145.

First[RealDigits[N[Gamma[1/6]/(2Gamma[1/3]Gamma[5/6]),88]]]

1/(2*gamma(1/3)*gamma(5/6)/gamma(1/6)) \\ Michel Marcus, Sep 24 2022

Equals 1/(2*A081760) = A175379/(2*A073005*A203145).
Equals Sum_{k,m in Z^2} exp(-Pi*(2/sqrt(3))*(k^2+k*m+m^2))*exp(2*Pi*i*(k/3-m/3)).
Equals Sum_{k>=0} (binomial(-1/3,2*k)^2 - binomial(-1/3,2*k+1)^2). - Gerry Martens, Jul 24 2023
Equals 3*Gamma(1/3)^3 / (2^(8/3) * Pi^2). - Vaclav Kotesovec, Jul 27 2023

A355178 Decimal expansion of 2^(-2/3)/L, where L is the conjectured Landau's constant A081760.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Sep 23 2022

			1.159595266963928365769992051570020881945...

Markus Faulhuber, An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a "Weltkonstante"-or-how Ramanujan split temperatures, The Ramanujan Journal volume 54, pages 1-27 (2021). See p. 23.

Cf. A002580, A073005, A081760, A175379, A203145, A235362, A357318.

First[RealDigits[N[2^(1/3)*Gamma[1/6]/(2Gamma[1/3]Gamma[5/6]), 90]]]

Equals A002580*A357318.
Equals A235362/A081760 = A235362*A175379/(A073005*A203145).
Equals Sum_{k,m in Z^2} exp(-Pi*(2/sqrt(3))*(k^2+k*m+m^2)).
From Gerry Martens, Jul 29 2023: (Start)
Equals hypergeom([1/3, 2/3], [1], 1/2).
Equals sqrt(Pi)/(Gamma(2/3)*Gamma(5/6)). (End)

cp's OEIS Frontend

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

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A246745 Decimal expansion of Gamma(2/5).

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

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A034789 Related to sextic factorial numbers A008542.

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Magma

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A091546 First column of the array A092077 ((8,2)-Stirling2).

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A243308 Decimal expansion of h_3, a constant related to certain evaluations of the gamma function from elliptic integrals.

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Maple

Mathematica

Formula

A357318 Decimal expansion of 1/(2*L), where L is the conjectured Landau's constant A081760.

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