A068466 -id:A068466 - cp's OEIS frontend

A240964 Decimal expansion of Sum_{n>=1} n/sinh(n*Pi).

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

Offset: 0

Jean-François Alcover, Aug 05 2014

Prudnikov (p. 721, section 5.3.5, formula 1) has a typo, Gamma(1/4)^4 is correct, not Gamma(1/4)^2. - Vaclav Kotesovec, May 19 2022

			0.09457301966476193951358890085441384931495532931922401...

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1 (Overseas Publishers Association, Amsterdam, 1986).

Cf. A068466, A335414, A335415.

Join[{0}, RealDigits[Gamma[1/4]^4/(32*Pi^3) - 1/(4*Pi), 10, 100] // First]
N[EllipticK[k]/Pi^2*(EllipticK[k] - EllipticE[k]) /. k -> 1/2, 100] (* Vaclav Kotesovec, May 19 2022 *)

suminf(k=1, k/sinh(k*Pi)) \\ Vaclav Kotesovec, May 19 2022

suminf(k=1, 1/(2*sinh((k - 1/2)*Pi)^2)) \\ Vaclav Kotesovec, May 19 2022

Gamma(1/4)^4/(32*Pi^3) - 1/(4*Pi).

A249205 Decimal expansion of the logarithmic capacity of the unit disk.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 23 2014

			0.59017029950804811302266897027924429361685831744...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.9 Integer Chebyshev constants, p. 268.

Steven Finch, Electrical capacitance [Cached copy, with permission of the author]

Cf. A068466, A249206, A249220.

k = (1/(4*Pi^(3/2)))*Gamma[1/4]^2; RealDigits[k, 10, 100] // First

(1/(4*Pi^(3/2)))*gamma(1/4)^2 \\ Michel Marcus, Sep 03 2023

Equals (1/(4*Pi^(3/2)))*Gamma(1/4)^2.

Equals hypergeom([1/2, 1/2], [1], 1/2)/2. - Gerry Martens, Jul 31 2023

A235136 a(n) = (2n - 1) a(n-2) for n>1, a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 3, 5, 21, 45, 231, 585, 3465, 9945, 65835, 208845, 1514205, 5221125, 40883535, 151412625, 1267389585, 4996616625, 44358635475, 184874815125, 1729986783525, 7579867420125, 74389431691575, 341094033905625, 3496303289504025, 16713607661375625

Offset: 0

Michael Somos, Jan 03 2014

			G.f. = 1 + x + 3*x^2 + 5*x^3 + 21*x^4 + 45*x^5 + 231*x^6 + 585*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..730

Cf. A007662, A007696, A008545, A196265.

Cf. A068465, A068466.

a[ n_] := 2^n If[ OddQ[n], 2 Pochhammer[ 1/4, (n + 1)/2], Pochhammer[ 3/4, n/2]]; (* Michael Somos, Jan 16 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (-2 Gamma[5/2] HermiteH[ -3/2, x] + (3 Gamma[5/4] + 2 Gamma[7/4]) Hypergeometric1F1[ 3/4, 1/2, x^2]) / (3 Gamma[5/4]), {x, 0, n}] // FullSimplify]; (* Michael Somos, Jan 16 2014 *)
RecurrenceTable[{a[0]==a[1]==1, a[n]==(2 n - 1) a[n - 2]}, a, {n, 25}] (* Vincenzo Librandi, Aug 08 2018 *)

{a(n) = if( n<0, (-1)^(-n\2) / a(-1-n), if( n<2, 1, (2*n - 1) * a(n-2)))};

Let b(n) = a(2*n - 2) / a(2*n + 1). Then b(-n) = b(n), 0 = b(n+1) * (b(n+1) + 2*b(n+2)) + b(n) * (2*b(n+1) - 5*b(n+2)) for all n in Z.
a(n-1) + a(n-2) = A196265(n) if n>1.
a(2*n) = A008545(n). a(2*n - 1) = A007696(n). a(n) = A007662(2*n - 1).
E.g.f. A(x) =: y satisfies 0 = y * 3 + y' * 2*x - y''.
0 = a(n)*(2*a(n+1) - a(n+3)) + a(n+1)*(a(n+2)) for all n in Z. - Michael Somos, Jan 24 2014
Let b(n) = a(n - 2) / a(n + 1). Then b(-n) = (-1)^n * b(n), 0 = b(n) * (b(n+1) - 4*b(n+3)) + b(n+2) * (2*b(n+1) + b(n+3)) for all n in Z. - Michael Somos, Sep 13 2014
a(n) ~ c * sqrt(Pi) * (2*n)^(n/2+1/4) / exp(n/2), where c = 2/Gamma(1/4) if n is odd, and 1/Gamma(3/4) if n is even. - Amiram Eldar, Sep 01 2025

A071002 Binary expansion of Gamma(1/4).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1

Offset: 2

Benoit Cloitre, May 18 2002

			11.1010000...

Cf. A068466, A071001.

RealDigits[Gamma[1/4], 2, 100][[1]] (* Amiram Eldar, May 04 2022 *)

Two 1's added in front and offset corrected by R. J. Mathar, Feb 05 2009

A078127 Decimal expansion of DirichletBeta'(1).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Nov 19 2002

			0.1929013167969124293631897640...

Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 8.
Eric Weisstein's World of Mathematics, Dirichlet Beta Function

Cf. A000796, A001620, A068465, A068466.

Pi/4*(gamma+log(2*Pi)-2*log(GAMMA(1/4)/GAMMA(3/4))); evalf(%) ; # R. J. Mathar, Jun 10 2024

Prepend@@RealDigits[(Pi*(EulerGamma + 2*Log[2] + 3*Log[Pi] - 4*Log[Gamma[1/4]]))/4, 10, 101]

Equals (Pi/4)*(gamma + log(2*Pi) - 2*log(Gamma(1/4)/Gamma(3/4))), where gamma is Euler's constant and Gamma(x) is the Euler Gamma function.

Equals Sum_{k>=1} (-1)^(k+1)*log(2*k+1)/(2*k+1). - Jean-François Alcover, Aug 11 2014

A241017 Decimal expansion of Sierpiński's S constant, which appears in a series involving the function r(n), defined as the number of representations of the positive integer n as a sum of two squares. This S constant is the usual Sierpiński K constant divided by Pi.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 08 2014

			0.822825249678847032995328716261464949475693118894850218393815613...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.10 Sierpinski's Constant, p. 123.

M. W. Coffey, Summatory relations and prime products for the Stieltjes constants, and other related results, arXiv:1701.07064 (2017) Proposition 9.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 103.
Guillaume Melquiond, W. Georg Nowak, Paul Zimmermann, Numerical approximation of the Masser-Gramain constant to four decimal places, Mathematics of Computation, Volume 82, Number 282, April 2013, Pages 1235-1246
Eric Weisstein's MathWorld, Sierpiński's Constant
Wikipedia, Sierpiński's Constant

Cf. A062089, A062539, A068466, A086058.

S = Log[4*Pi^3*Exp[2*EulerGamma]/Gamma[1/4]^4]; RealDigits[S, 10, 104] // First

log(agm(sqrt(2), 1)^2/2) + 2*Euler \\ Charles R Greathouse IV, Nov 26 2024

S = gamma + beta'(1) / beta(1), where beta is Dirichlet's beta function.
S = log(Pi^2*exp(2*gamma) / (2*L^2)), where L is Gauss' lemniscate constant.
S = log(4*Pi^3*exp(2*gamma) / Gamma(1/4)^4), where gamma is Euler's constant and Gamma is Euler's Gamma function.
S = A062089 / Pi, where A062089 is Sierpiński's K constant.
S = A086058 - 1, where A086058 is the conjectured (but erroneous!) value of Masser-Gramain 'delta' constant. [updated by Vaclav Kotesovec, Apr 27 2015]
S = 2*gamma + (4/Pi)*integral_{x>0} exp(-x)*log(x)/(1-exp(-2*x)) dx.
Sum_{k=1..n} r(k)/k = Pi*(log(n) + S) + O(n^(-1/2)).
Equals 2*A001620 - A088538*A115252 [Coffey]. - R. J. Mathar, Jan 15 2021

A323755 Decimal expansion of a constant related to the asymptotics of A203475.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jan 26 2019

			0.274528350333552903800408993482507428142383783773190451181072723742691...

Vaclav Kotesovec, Table of n, a(n) for n = 0..296

Cf. A068466, A203475.

RealDigits[Sqrt[Gamma[1/4]] * E^(Pi/24) / (2^(9/8) * Pi^(9/8)), 10, 120][[1]]

Equals limit_{n->infinity} A203475(n) / (2^(n^2/2) * exp(Pi*n*(n+1)/4 - 3*n^2/2 + n) * n^(n*(n-1) - 3/4)).

Equals sqrt(Gamma(1/4)) * exp(Pi/24) / (2*Pi)^(9/8).

A068153 Continued fraction for Gamma(1/4).

Original entry on oeis.org

3, 1, 1, 1, 2, 25, 4, 9, 1, 1, 8, 4, 1, 6, 1, 1, 19, 1, 1, 4, 1, 1, 5, 5, 7, 4, 1, 7, 1, 4, 2, 1, 4, 7, 1, 6, 4, 1, 4, 14, 1, 2, 24, 3, 25, 8, 3, 1, 1, 1, 1, 4, 1, 1, 28, 1, 1, 1, 12, 1, 1, 23, 1, 2, 2, 1, 3, 1, 1, 1, 6, 1, 5, 16, 8, 6, 4, 8, 2, 9, 3, 1, 4, 75, 5, 4, 5, 1, 1, 3, 2, 1, 1, 12, 2, 1, 2, 6

Offset: 0

Benoit Cloitre, Mar 13 2002

			gamma(1/4) = 3.62560990822190... = 3 + 1/(1 + 1/(1 + 1/(1 + 1/(2 + ...)))). - _Harry J. Smith_, Apr 20 2009

Harry J. Smith, Table of n, a(n) for n = 0..999

Cf. A068466 (decimal expansion).

{ default(realprecision, 1080); x=contfrac(gamma(1/4)); for (n=1, 1000, write("b068153.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 20 2009

Offset changed by Andrew Howroyd, Aug 07 2024

A196535 Decimal expansion of Sum_{j=0..oo} exp(-Pi(2j+1)^2).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 03 2011

			0.04321391826429779829201838202725...

Jolley, Summation of Series, Dover (1961) eq (114) on page 22.
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1 (Overseas Publishers Association, Amsterdam, 1986), p. 729, formula 14.

Michael I. Shamos, A catalog of the real numbers, (2007), p. 78.

Cf. A093580, A068466, A010767.

(root[4](2)-1)*GAMMA(1/4)/2^(11/4)/Pi^(3/4) ; evalf(%) ;

RealDigits[ EllipticTheta[2, 0, Exp[-4*Pi]]/2, 10, 105] // First // Prepend[#, 0]&  (* Jean-François Alcover, Feb 12 2013 *)

Equals (2^(1/4)-1) * Gamma(1/4) / ( 2^(11/4) * Pi^(3/4) ).

Equals theta2(exp(-4*Pi))/2.

12 more digits from Jean-François Alcover, Feb 12 2013

A256717 Decimal expansion of G(5/4) where G is the Barnes G-function.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 09 2015

			1.0650445385309557171597175836949771419373490732697618922213...

G. C. Greubel, Table of n, a(n) for n = 1..10000
J.-P. Allouche, A note on products involving zeta(3) and Catalan's constant. arXiv:1305.6247v3 [math.NT], 2013-2014, p. 7.
Eric Weisstein's MathWorld, Barnes G-Function
Wikipedia, Barnes G-function

Cf. A006752 (Catalan), A068466 (Gamma(1/4)), A074962 (Glaisher), A087013 (G(1/4)), A087014 (G(1/2)), A087015 (G(3/4)), A087016 (G(3/2)), A087017 (G(5/2)).

RealDigits[BarnesG[5/4], 10, 104] // First
RealDigits[Exp[3/32 - Catalan/(4*Pi)]*Gamma[1/4]^(1/4)/Glaisher^(9/8), 10, 100][[1]] (* G. C. Greubel, Aug 25 2018 *)

exp(3/32 - Catalan/(4*Pi))*gamma(1/4)^(1/4)/exp(3/32-9*zeta'(-1)/8) \\ Charles R Greathouse IV, Jul 01 2016

Equals exp(3/32 - Catalan/(4*Pi))*Gamma(1/4)^(1/4)/Glaisher^(9/8).

Equals G(1/4)*Gamma(1/4). - Vaclav Kotesovec, Apr 09 2015

cp's OEIS Frontend

A240964 Decimal expansion of Sum_{n>=1} n/sinh(n*Pi).

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A249205 Decimal expansion of the logarithmic capacity of the unit disk.

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A235136 a(n) = (2*n - 1) * a(n-2) for n>1, a(0) = a(1) = 1.

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A071002 Binary expansion of Gamma(1/4).

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A078127 Decimal expansion of DirichletBeta'(1).

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A241017 Decimal expansion of Sierpiński's S constant, which appears in a series involving the function r(n), defined as the number of representations of the positive integer n as a sum of two squares. This S constant is the usual Sierpiński K constant divided by Pi.

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A323755 Decimal expansion of a constant related to the asymptotics of A203475.

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A068153 Continued fraction for Gamma(1/4).

A235136 a(n) = (2n - 1) a(n-2) for n>1, a(0) = a(1) = 1.

A196535 Decimal expansion of Sum_{j=0..oo} exp(-Pi(2j+1)^2).