A068466 -id:A068466 - cp's OEIS frontend

A269545 Decimal expansion of Gamma(Pi).

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

Offset: 1

Iaroslav V. Blagouchine, Feb 29 2016

			2.2880377953400324179595889090602339228896881533562224...

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

Cf. A269546, A269547, A269557, A269558, A269559, A257955, A257957, A257958, A257959, A002161, A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

MATLAB
```
format long; gamma(pi)
```
Maple
```
evalf(GAMMA(Pi), 120);
```
Mathematica
```
RealDigits[Gamma[Pi], 10, 120][[1]]
```
PARI
```
default(realprecision, 120); gamma(Pi)
```

Equals Integral_{x >= 0} x^(Pi-1)/e^x dx (Euler integral of the second kind).

A269546 Decimal expansion of log(Gamma(Pi)).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Feb 29 2016

Gamma(x) is the Gamma function (Euler's integral of the second kind).

			0.8276945923234371015295785584523599511535017341207373...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A269545, A269547, A269557, A269558, A269559, A257955, A257957, A257958, A257959, A002161.

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

MATLAB
```
format long; log(gamma(pi))
```
Maple
```
evalf(lnGAMMA(Pi), 120);
```
Mathematica
```
RealDigits[LogGamma[Pi], 10, 120][[1]]
```

default(realprecision, 120); lngamma(Pi)

A269547 Decimal expansion of Psi(Pi).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Feb 29 2016

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

			0.9772133079420067332920694864061823436408346099943256...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A269545, A269546, A269557, A269558, A269559, A257955, A257957, A257958, A257959, A002161.

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

MATLAB
```
format long; psi(pi)
```
Maple
```
evalf(Psi(Pi), 120)
```
Mathematica
```
RealDigits[PolyGamma[Pi], 10, 120][[1]]
```
PARI
```
default(realprecision, 120); psi(Pi)
```

A269557 Decimal expansion of Gamma(log(2)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Feb 29 2016

Gamma(x) is the Gamma function (Euler's integral of the second kind).

			1.3090409112814812698245325213959295756125890319181890...

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

Cf. A269558, A269559, 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; gamma(log(2))
```
Maple
```
evalf(GAMMA(ln(2)), 120);
```
Mathematica
```
RealDigits[Gamma[Log[2]], 10, 120][[1]]
```

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

A269558 Decimal expansion of log(Gamma(log(2))).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Feb 29 2016

Gamma(x) is the Gamma function (Euler's integral of the second kind).

			0.2692947402831312429499165832117128248889035102111661...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A269557, A269559, 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; log(gamma(log(2)))
```
Maple
```
evalf(lnGAMMA(ln(2)), 120);
```

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

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

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

Original entry on oeis.org

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))

A034255 Related to quartic factorial numbers A007696.

Original entry on oeis.org

1, 10, 120, 1560, 21216, 297024, 4243200, 61526400, 902387200, 13355330560, 199115837440, 2986737561600, 45030812467200, 681895160217600, 10364806435307520, 158063298138439680, 2417438677411430400, 37067393053641932800, 569667303771760230400, 8772876478085107548160

Offset: 1

Wolfdieter Lang

Michael De Vlieger, Table of n, a(n) for n = 1..833
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.
Index entries for sequences related to factorial numbers.

First column of triangle A048882.

Cf. A007696, A068466.

Rest[CoefficientList[Series[(-1+(1-16x)^(-1/4))/4,{x,0,20}],x]] (* Harvey P. Dale, May 19 2011 *)

a(n) = 4^(n-1)*A007696(n)/n!, where A007696(n) = (4*n-3)(!^4) = Product_{j=1..n} (4*j-3), n >= 1.
G.f.: (-1+(1-16*x)^(-1/4))/4.
a(n) = A048882(n, 1).
Convolution of A034385(n-1) with A025749(n), n >= 1.
D-finite with recurrence: n*a(n) + 4*(-4*n+3)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
a(n) ~ 2^(4*n-2) * n^(-3/4) / Gamma(1/4). - Amiram Eldar, Aug 18 2025

A068464 Factorial expansion of Gamma(1/4) = Sum_{n>=1} a(n)/n! with largest possible a(n), and Gamma = Euler's gamma function.

Original entry on oeis.org

3, 1, 0, 3, 0, 0, 3, 0, 5, 3, 2, 7, 0, 2, 8, 9, 16, 3, 1, 15, 18, 8, 20, 7, 23, 8, 10, 11, 28, 29, 24, 30, 3, 16, 10, 8, 31, 11, 30, 35, 5, 5, 38, 32, 31, 42, 13, 17, 43, 3, 41, 27, 1, 14, 26, 52, 38, 22, 55, 46, 6, 35, 46, 34, 24, 52, 52, 64, 62, 25, 46, 56, 3, 71, 70, 22, 53, 63, 53

Offset: 1

Benoit Cloitre, Mar 10 2002

			Gamma(1/4) = A068466 = 3.6256099... = 3/1! + 1/2! + 0 + 3/4! + 0 + 0 + 3/7! + 0 + 5/9! + 3/10! + 2/11! + ... - _M. F. Hasler_, Nov 26 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for factorial base representation

Cf. A007514, A068466 (decimal expansion), A068463.

SetDefaultRealField(RealField(250));  [Floor(Gamma(1/4))] cat [Floor(Factorial(n)*Gamma(1/4)) - n*Floor(Factorial((n-1))*Gamma(1/4)) : n in [2..80]]; // G. C. Greubel, Nov 27 2018

r:= Gamma[1/4]; Table[If[n == 1, Floor[r], Floor[n!*r]- n*Floor[(n-1)!*r] ], {n,1,100}] (* G. C. Greubel, Mar 29 2018 *)

default(realprecision, 250); b = gamma(1/4); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Mar 29 2018

A068464(N=90,c=gamma(precision(.25,logint(N!,10)+1)))=vector(N,n,if(n>1,c=c%1*n,c)\1) \\ - M. F. Hasler, Nov 26 2018

def A068464(n):
    if (n==1): return floor(gamma(1/4))
    else: return expand(floor(factorial(n)*gamma(1/4)) - n*floor(factorial(n-1)*gamma(1/4)))
[A068464(n) for n in (1..80)] # G. C. Greubel, Nov 27 2018

a(n) = floor(n!*Gamma(1/4)) - n*floor((n-1)!*Gamma(1/4)), for n > 1. - M. F. Hasler, Nov 26 2018

A225119 Decimal expansion of Integral_{x=0..Pi/2} sin(x)^(3/2) dx.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 29 2013

			0.87401918476403993682161319663037313789425165047720772093894...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 195.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant p. 102 and Section 6.1 Gauss' Lemniscate Constant p. 422.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A062539, A068466, A093341, A085565.

evalf((1/3)*sqrt(2)*EllipticK(1/sqrt(2)), 120); # Vaclav Kotesovec, Apr 22 2015

RealDigits[1/3*Sqrt[2]*EllipticK[1/2], 10, 100][[1]]

sqrt(Pi)*gamma(1/4)/(6*gamma(3/4)) \\ G. C. Greubel, Apr 01 2017

ellK(sqrt(1/2))*sqrt(2)/3 \\ Charles R Greathouse IV, Feb 04 2025

Equals 1/3 * sqrt(2) * ellipticK(1/2), (defined as in Mathematica).
Equals sqrt(2)/6 * Pi * hypergeom([1/2,1/2],[1],1/2).
Equals gamma(1/4)^2/(6*sqrt(2*Pi)).
Equals sqrt(Pi)*gamma(1/4)/(6*gamma(3/4)).
Equals Integral_{0..1} (1-x^2)^(1/4) dx.
Equals Integral_{0..1} sqrt(1-x^4) dx. - Charles R Greathouse IV, Aug 21 2017
Equals (2/3)*A085565. - Peter Bala, Oct 27 2019
Equals A062539/3. - Hugo Pfoertner, Dec 15 2024

A203144 Decimal expansion of Gamma(5/8).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 29 2011

			1.4345188480905567756360197394564231366322077722066673307706...

Index to sequences related to the Gamma function

RealDigits[Gamma[5/8],10,120][[1]] (* Harvey P. Dale, Sep 14 2018 *)

gamma(5/8) \\ Charles R Greathouse IV, Mar 25 2014

this * A203142 = 2^(3/4)*sqrt(Pi) * A068466. - R. J. Mathar, Jan 15 2021

cp's OEIS Frontend

A269545 Decimal expansion of Gamma(Pi).

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A269546 Decimal expansion of log(Gamma(Pi)).

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A269547 Decimal expansion of Psi(Pi).

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A269557 Decimal expansion of Gamma(log(2)).

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A269558 Decimal expansion of log(Gamma(log(2))).

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MATLAB

Maple

Mathematica

PARI

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

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MATLAB

Maple

Mathematica

PARI

A034255 Related to quartic factorial numbers A007696.