A203140 -id:A203140 - cp's OEIS frontend

A256066 Decimal expansion of log(Gamma(1/12)).

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

Offset: 1

Iaroslav V. Blagouchine, Mar 18 2015

			2.44229731118288975091554935219408858208684110709150...

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

Cf. A203140 (Gamma(1/12)), A256165 (log(Gamma(1/3))), A256166 (log(Gamma(1/4))), A256167 (log(Gamma(1/5))), A255888 (log(Gamma(1/6))), A255306 (log(Gamma(1/8))), A255189 (first generalized Stieltjes constant at 1/12, gamma_1(1/12)).

evalf(log(GAMMA(1/12)),100);
evalf(-(1/4)*log(2)+(3/8)*log(3)+(1/2)*log(1+sqrt(3))-(1/2)*log(Pi)+log(GAMMA(1/4))+log(GAMMA(1/3)), 100);

RealDigits[Log[Gamma[1/12]],10,100][[1]]

PARI
```
log(gamma(1/12))
```

Equals -(1/4)*log(2) + (3/8)*log(3) + (1/2)*log(1+sqrt(3)) - (1/2)*log(Pi) + log(Gamma(1/4)) + log(Gamma(1/3)).

A257955 Decimal expansion of Gamma(1/Pi).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, May 14 2015

The reference gives an interesting product representation in terms of rational multiple of 1/Pi for Gamma(1/Pi).

			2.8112975146708616421227908037104816935281655223291765...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/Pi, Mathematics of Computation (AMS), 2015.

Cf. A257957, A257958, A257959, A002161, A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

Maple
```
evalf(GAMMA(1/Pi), 117);
```
Mathematica
```
RealDigits[Gamma[1/Pi], 10, 117][[1]]
```

default(realprecision, 117); gamma(1/Pi)

A269545 Decimal expansion of Gamma(Pi).

Original entry on oeis.org

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))

A020004 Nearest integer to Gamma(n + 1/12)/Gamma(1/12).

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 12, 73, 519, 4193, 38084, 384010, 4256112, 51428023, 672849973, 9475970455, 142929221024, 2298778304796, 39270796040273, 710146895061598, 13551969914092152, 272168729108017393, 5738224038694033364

Offset: 0

Simon Plouffe

nonn

			Gamma(0 + 1/12)/Gamma(1/12) = 1, so a(0) = 1.
Gamma(1 + 1/12)/Gamma(1/12) = 1/12 = 0.08333..., so a(1) = 0.
Gamma(2 + 1/12)/Gamma(1/12) = 13/144 < 1/2, so a(2) = 0.
Gamma(3 + 1/12)/Gamma(1/12) = 325/1728 < 1/2, so a(3) = 0.
Gamma(4 + 1/12)/Gamma(1/12) = 12025/20736 = 0.5799..., so a(4) = 1.
Gamma(5 + 1/12)/Gamma(1/12) = 589225/248832 = 2.3679631237..., so a(5) = 2.
Gamma(6 + 1/12)/Gamma(1/12) = 35942725/2985984 = 12.037145878879458..., so a(6) = 12.
Gamma(7 + 1/12)/Gamma(1/12) = 2623818925/35831808 = 73.22597..., so a(7) = 73.

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A020049, A020094, A021016 (decimal expansion of 1/12), A203140 (decimal expansion of Gamma(1/12)).

[Round(Gamma(n +1/12)/Gamma(1/12)): n in [0..30]]; // G. C. Greubel, Jan 19 2018

Digits := 64:f := proc(n,x) round(GAMMA(n+x)/GAMMA(x)); end;

Table[Round[Gamma[n + 1/12]/Gamma[1/12]], {n, 0,50}] (* G. C. Greubel, Jan 19 2018 *)

for(n=0,30, print1(round(gamma(n+1/12)/gamma(1/12)), ", ")) \\ G. C. Greubel, Jan 19 2018

A272097 Decimal expansion of an infinite product involving the ratio of n! to its Stirling approximation.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Apr 20 2016

Product_{k=1..n} (k! / (sqrt(2*Pi*k) * k^k * exp(-k))) ~ c * n^(1/12), where c = exp(1/12)*(2*Pi)^(1/4) / A^2 = A213080 = 1.04633506677050318098095065697776..., where A = A074962 is the Glaisher-Kinkelin constant.

Product_{n>=1} (n! / (sqrt(2*Pi*n) * n^n * exp(-n) * (1 + 1/(12*n) + 1/(288*n^2)))) = exp(1/12) * (2*Pi)^(1/4) * abs(Gamma(25/24 + i/24))^2 / A^2 = 0.997305599490607358564533726617761207426462854447669845..., where A = A074962 is the Glaisher-Kinkelin constant and i is the imaginary unit.

			1.00268791324152794158434554643452096181810403192367888372866567380647785...

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

Cf. A000142, A001163, A001164, A074962, A203138, A203140, A213080, A241140.

Product[n!/(n^n/E^n*Sqrt[2*Pi*n]*(1 + 1/(12*n))), {n, 1, Infinity}]
RealDigits[E^(1/12)*(2*Pi)^(1/4)*Gamma[13/12]/Glaisher^2, 10, 120][[1]]

Product_{n>=1} (n! / (sqrt(2*Pi*n) * n^n * exp(-n) * (1 + 1/(12*n)))).

Equals exp(1/12) * (2*Pi)^(1/4) * Gamma(1/12) / (12 * A^2), where A = A074962 is the Glaisher-Kinkelin constant.

cp's OEIS Frontend

A256066 Decimal expansion of log(Gamma(1/12)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A257955 Decimal expansion of Gamma(1/Pi).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A269545 Decimal expansion of Gamma(Pi).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

MATLAB

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Maple

Mathematica

PARI

A269547 Decimal expansion of Psi(Pi).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Maple

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Maple

Mathematica

PARI

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

Views

Author