A200134 -id:A200134 - cp's OEIS frontend

A020777 Decimal expansion of (-1)*Gamma'(1/4)/Gamma(1/4) where Gamma(x) denotes the Gamma function.

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

Offset: 1

Benoit Cloitre, May 24 2003

			4.2274535333762654080895301460966835773672444387082422716552795595189567958...

S.J. Patterson, "An introduction to the theory of the Riemann zeta function", Cambridge studies in advanced mathematics no. 14, p. 135, 1995.

G. C. Greubel, Table of n, a(n) for n = 1..10000
E. D. Krupnikov, K. S. Kölbig, Some special cases of the generalized hypergeometric function (q+1)Fq, J. Comp. Appl. Math. 78 (1997) 79-95, psi(1/4).
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A001620, A200134, A301816.

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R) + Pi(R)/2 + Log(8); // G. C. Greubel, Aug 28 2018

evalf(gamma+3*log(2)+Pi/2) ; # R. J. Mathar, Nov 13 2011
evalf(abs(Psi(1/4))) ; # R. J. Mathar, Nov 19 2024

EulerGamma + Pi/2 + Log[8] // RealDigits[#, 10, 105][[1]] & (* Jean-François Alcover, Jun 18 2013 *)
N[StieltjesGamma[0, 1/4], 99] (* Peter Luschny, May 16 2018 *)

PARI
```
Euler+3*log(2)+Pi/2
```

Gamma'(1/4)/Gamma(1/4) = -EulerGamma - 3*log(2) - Pi/2 where EulerGamma is the Euler-Mascheroni constant (A001620).

Pi = gamma(0,1/4) - gamma(0,3/4) = A020777 - A200134, where gamma(n,x) denotes the generalized Stieltjes constants. - Peter Luschny, May 16 2018

A200135 Decimal expansion of the negated value of the digamma function at 1/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(1/5) =  -5.289039896592188295547207962...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200064, A200134, A200136, A200137, A200138.

SetDefaultRealField(RealField(100)); R:= RealField(); -EulerGamma(R) -Pi(R)*Sqrt(1+2/Sqrt(5))/2 -5*Log(5)/4 -Sqrt(5)/4*Log((3+Sqrt(5)/2) ); // G. C. Greubel, Sep 03 2018

-gamma-Pi*sqrt(1+2/sqrt(5))/2-5*log(5)/4-sqrt(5)/4*log((3+sqrt(5)/2) ); evalf(%) ;

RealDigits[-PolyGamma[1/5], 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

-psi(1/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(1/5) = -gamma - Pi*sqrt(1 + 2/sqrt(5))/2 - 5*log(5)/4 -sqrt(5)*log((3 + sqrt(5))/2)/4 where gamma = A001620, sqrt(1 + 2/sqrt(5)) = A019952, (3 + sqrt(5))/2 = A104457.

More terms from Jean-François Alcover, Feb 11 2013

A200138 Decimal expansion of the negated value of the digamma function at 4/5.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Nov 13 2011

			Psi(4/5) = -0.965008566706138459391297633...

Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200134-A200137.

-gamma+Pi*sqrt(1+2/sqrt(5))/2-5*log(5)/4-sqrt(5)/4*log(3/2+sqrt(5)/2) ; evalf(%) ;

RealDigits[ -PolyGamma[4/5], 10, 87] // First (* Jean-François Alcover, Feb 20 2013 *)

-psi(4/5) \\ Charles R Greathouse IV, Nov 22 2011

Psi(4/5) = -gamma + Pi*sqrt(1+2/sqrt 5)/2 -5*log(5)*log((3+sqrt 5)/2)/4.

A074638 Denominator of 1/3 + 1/7 + 1/11 + ... + 1/(4n-1).

Original entry on oeis.org

3, 21, 231, 385, 7315, 168245, 4542615, 140821065, 28164213, 366134769, 15743795067, 739958368149, 12579292258533, 62896461292665, 3710891216267235, 3710891216267235, 248629711489904745, 17652709515783236895, 88263547578916184475, 6972820258734378573525

Offset: 1

Robert G. Wilson v, Aug 27 2002

This s(n) := Sum_{j=0..n-1} 1/(4*j + 3), for n >= 1, equals (Psi(n + 3/4) - Psi(3/4))/4, with the digamma function Psi(z). See Abramowitz-Stegun, p. 258, eqs. 6.3.7 and 6.3.5, with z -> 3/4. A200134 = -Psi(3/4). - Wolfdieter Lang, Apr 06 2022

Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions. p.258, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. p. 258.

The numerators times 4 are A074637.

Cf. A075135, A200134.

Table[ Denominator[ Sum[1/i, {i, 3/4, n}]], {n, 1, 20}]

a(n) = denominator(sum(i=1, n, 1/(4*i-1))); \\ Michel Marcus, Mar 21 2021

from fractions import Fraction
def a(n): return sum(Fraction(1, 4*i-1) for i in range(1, n+1)).denominator
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Mar 21 2021

Denominator( (Psi(n + 3/4) - Psi(3/4))/4 ). See the comment above. - Wolfdieter Lang, Apr 05 2022

A200137 Decimal expansion of the negated digamma function at 3/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(3/5) = -1.540619213893190414760663948806231941510...

Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200134-A200138

-gamma+Pi*sqrt(1-2/sqrt(5))/2-5*log(5)/4+sqrt(5)/4*log(3/2+sqrt(5)/2) ; evalf(%) ;

RealDigits[-PolyGamma[3/5], 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

-psi(3/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(3/5) = -gamma +Pi*sqrt( 1-2/sqrt 5)/2 -5*log(5)/4 +sqrt(5)*log((3+sqrt 5)/2)/4.

More terms from Jean-François Alcover, Feb 11 2013

A352395 Denominator of Sum_{k=0..n} (-1)^k / (2*k+1).

Original entry on oeis.org

1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675, 13835020108241056725

Offset: 0

Wolfdieter Lang, Apr 06 2022

This is not the sequence A025547(n+1)_{n>=0}, because a(32) = 1420993851085122917681925 but A025547(33) = 18472920064106597929865025. Hence it is also not the sequence A350670.
The alternating sum Sum_{k=0..n} (-1)^k/(2*k+1) = (Psi(n + 3/2) - Psi((2*n - (-1)^n)/4 + 1) - log(2) + Pi/2)/2, with the Digamma function Psi(z).
Proof by subtracting twice the negative fractions from Sum_{k=0..n} 1/(2*k+1) = A350669(n)/A350670(n) (Abramowitz-Stegun, p. 258, eq. 6.3.4.), using Sum_{j=0..k} 1/(4*j + 3) = A074637((k+1)/4)/A074638(k+1) (Abramowitz-Stegun, p. 258, eqs. 6.3.6. with 6.3.5.) and, finally, replacing in the results for the even and odd n cases the formula for Psi(3/4) = -A200134.

Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions. p.258, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. p. 258.

Cf. A007509 (numerators).

Cf. A025547, A074637, A074638, A200134, A350669, A350670.

Denominator @ Accumulate @ Table[(-1)^k/(2*k + 1), {k, 0, 25}] (* Amiram Eldar, Apr 08 2022 *)

a(n) = denominator(sum(k=0, n, (-1)^k / (2*k+1))); \\ Michel Marcus, Apr 07 2022

from fractions import Fraction
def A352395(n): return sum(Fraction(-1 if k % 2 else 1,2*k+1) for k in range(n+1)).denominator # Chai Wah Wu, May 18 2022

a(n) = denominator( (Psi(n + 3/2) - Psi((2*n - (-1)^n)/4 + 1) - log(2) + Pi/2)/2), for n >= 0, with the Digamma function. See the above comment.

a(n) = denominator(Pi/4 + (-1)^n * (Psi((n + 5/2)/2) - Psi((n + 3/2)/2))/4). - Vaclav Kotesovec, May 16 2022

A375772 Decimal expansion of the absolute value of the second derivative of the digamma function at 3/4.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 27 2024

			psi''(3/4) = -5.3026332163376396314327...

Ernst D. Krupnikov and K. S. Kolbig, Some special cases of the generalized hypergeometric function _{q+1}F_q, J. Comp. Appl. Math. 78 (1997) 79-95.

Cf. A200134 (psi(3/4)), A282824 (psi'(3/4)).

Maple
```
2*(Pi^3-28*Zeta(3)); evalf(%) ;
```

RealDigits[PolyGamma[2, 3/4], 10, 105][[1]] (* Vaclav Kotesovec, Aug 27 2024 *)

psi''(3/4) = 2*(Pi^3-28*zeta(3)).

cp's OEIS Frontend

A020777 Decimal expansion of (-1)*Gamma'(1/4)/Gamma(1/4) where Gamma(x) denotes the Gamma function.

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A200135 Decimal expansion of the negated value of the digamma function at 1/5.

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A200138 Decimal expansion of the negated value of the digamma function at 4/5.

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A074638 Denominator of 1/3 + 1/7 + 1/11 + ... + 1/(4n-1).

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A200137 Decimal expansion of the negated digamma function at 3/5.

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Maple

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PARI

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A352395 Denominator of Sum_{k=0..n} (-1)^k / (2*k+1).

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A375772 Decimal expansion of the absolute value of the second derivative of the digamma function at 3/4.

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