A076788 -id:A076788 - cp's OEIS frontend

A233090 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number.

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

Offset: 0

Jean-François Alcover, Dec 04 2013, after the comment by Peter Bala about A233033

			0.7512855644747464283748363509446562442281164327128118011201697220886...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

R. Barbieri, J. A. Mignaco, and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864, table II (13)
Philippe Flajolet and Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998), page 32.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (C5.15)
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 561.

Cf. A002117 (zeta(3)), A197070 (3*zeta(3)/4), A233091 (7*zeta(3)/8), A076788 (alternating sum with denominator n), A152648 (non-alternating sum with denominator n^2), A152649 (non-alternating sum with denominator n^3), A233033 (alternating sum with denominator n^3).

RealDigits[ 5*Zeta[3]/8, 10, 100] // First

Equals 5*zeta(3)/8.
Equals -Integral_{x=0..1} (log(1+x)*log(1-x)/x)*dx. - Amiram Eldar, May 06 2023
Equals Sum_{m>=1} Sum_{n>=1} (-1)^(m-1)/(m*n*(m + n)) (see Finch). - Stefano Spezia, Nov 02 2024

A099218 Decimal expansion of Li_4(1/2).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Oct 06 2004

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), Section 1.3.2.

G. Rutledge, R. D. Douglass, Table of definite integrals, Am. Math. Monthly 45 (8) (1938) 525-530, eq. (22).

Cf. A076788, A099217, A099219, A099220, A099221, A099222, A099223, A099224.

polylog(4,1/2) ; evalf(%) ; # R. J. Mathar, Feb 15 2013

RealDigits[ N[ PolyLog[4, 1/2], 104]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

polylog(4,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_4(1/2)=sum(k>0, 1/2^k/k^4)=0.5174790616738993863...

Leading zero removed, formula value corrected by R. J. Mathar, Feb 05 2009

A099217 Decimal expansion of Li_3(1/2).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Oct 06 2004

			0.537213193608040200940623225594965826670402499340...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 44.
L. Lewin, Polylogarithms and associated Functions, North Holland (1981) A.2.6(3)

R. Barbieri, J. A. Mignaco, and E. Remiddi, Electron form factors up to fourth order. I, Il Nuovo Cim. 11A (4) (1972) 824-864, variable a_3 and table II (15).
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 473.

Cf. A076788, A099218, A099219, A099220, A099221, A099222, A099223, A099224.

RealDigits[PolyLog[3, 1/2], 10, 104] // First (* Jean-François Alcover, Feb 13 2013 *)

polylog(3,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_3(1/2) = Sum_{k>0} 1/(2^k*k^3) = 0.537213193608...

Li_3(1/2) = 7*zeta(3)/8-Pi^2*log(2)/12+log(2)^3/6. - Benoit Cloitre, May 22 2006

Leading zero removed by R. J. Mathar, Feb 05 2009

A099219 Decimal expansion of Li_5(1/2).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Oct 06 2004

			0.50840057924...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch (1996), Section 1.3.2.

Cf. A076788, A099217, A099218, A099220, A099221, A099222, A099223, A099224.

polylog(5,1/2) ; evalf(%) ; # R. J. Mathar, Feb 15 2013

RealDigits[N[PolyLog[5, 1/2], 104]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

polylog(5,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_5(1/2) = Sum_{k>0} 1/2^k/k^5.

A099220 Decimal expansion of Li_6(1/2).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Oct 06 2004

Cf. A076788, A099217, A099218, A099219, A099221, A099222, A099223, A099224.

RealDigits[ N[ PolyLog[6, 1/2], 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

polylog(6,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_6(1/2)=sum(k>0, 1/2^k/k^6)=0.5040953978...

A099221 Decimal expansion of Li_7(1/2).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Oct 06 2004

Cf. A076788, A099217, A099218, A099219, A099220, A099222, A099223, A099224.

RealDigits[ N[ PolyLog[7, 1/2], 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

polylog(7,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_7(1/2)=sum(k>0, 1/2^k/k^7)=0.50201456332470849456....

Formula value corrected and leading zero removed by R. J. Mathar, Feb 05 2009

A099222 Decimal expansion of Li_8(1/2).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Oct 06 2004

Cf. A076788, A099217, A099218, A099219, A099220, A099221, A099223, A099224.

RealDigits[ N[ PolyLog[8, 1/2], 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

polylog(8,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_8(1/2)=sum(k>0, 1/2^k/k^8)=0.500996659...

A099223 Decimal expansion of Li_9(1/2).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Oct 06 2004

Cf. A076788, A099217, A099218, A099219, A099220, A099221, A099222, A099224.

RealDigits[ N[ PolyLog[9, 1/2], 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

polylog(9,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_9(1/2)=sum(k>0, 1/2^k/k^9)=0.5004948881...

A099224 Decimal expansion of Li_{10}(1/2).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Oct 06 2004

Cf. A076788, A099217, A099218, A099219, A099220, A099221, A099222, A099223.

RealDigits[ N[ PolyLog[10, 1/2], 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

polylog(10,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_{10}(1/2)=sum(k>0, 1/2^k/k^10)=0.500246320606006775...

A274181 Decimal expansion of Phi(1/2, 2, 2), where Phi is the Lerch transcendent.

Original entry on oeis.org

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

Offset: 0

Johannes W. Meijer and N. H. G. Baken, Jun 17 2016, Jul 08 2016

The exponential integral distribution is defined by p(x, m, n, mu) = ((n+mu-1)^m * x^(mu-1) / (mu-1)!) * E(x, m, n), see A163931 and the Meijer link. The moment generating function of this probability distribution function is M(a, m, n, mu) = Sum_{k>=0}(((mu+k-1)!/((mu-1)!*k!)) * ((n+mu-1) / (n+mu+k-1))^m * a^k).

In the special case that mu=1 we get p(x, m, n, mu=1) = n^m * E(x, m, n) and M(a, m, n, mu=1) = n^m * Phi(a, m, n), with Phi the Lerch transcendent. If n=1 and mu=1 we get M(a, m, n=1, mu=1) = polylog(m, a)/a = Li_m(a)/a.

			0.32896210586005002361062528063872043497679389922...

William Feller, An introduction to probability theory and its applications, Vol. 1. p. 285, 1968.

J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No. 3, April 1987. pp 209-211.
Eric W. Weisstein’s World of Mathematics, Lerch transcendent.
Eric W. Weisstein’s World of Mathematics, Polylogarithm.

Cf. A163931, A002162 (Phi(1/2, 1, 1)/2), A076788 (Phi(1/2, 2, 1)/2), A112302, A008276.

Digits := 101; c := evalf(LerchPhi(1/2, 2, 2));

N[HurwitzLerchPhi[1/2, 2, 2], 25] (* G. C. Greubel, Jun 19 2016 *)

Pi^2/3 - 2*log(2)^2 - 2 \\ Altug Alkan, Jul 08 2016

lerchphi(.5,2,2) \\ Charles R Greathouse IV, Jan 30 2025

from mpmath import mp, lerchphi
mp.dps=102
print([int(d) for d in list(str(lerchphi(1/2, 2, 2))[2:-1])]) # Indranil Ghosh, Jul 04 2017

Equals Phi(1/2, 2, 2) with Phi the Lerch transcendent.
Equals Sum_{k>=0}(1/((2+k)^2*2^k)).
Equals 4 * polylog(2, 1/2) - 2.
Equals Pi^2/3 - 2*log(2)^2 - 2.
Equals Integral_{x=0..oo} x*exp(-x)/(exp(x)-1/2) dx. - Amiram Eldar, Aug 24 2020

cp's OEIS Frontend

A233090 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number.

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A099218 Decimal expansion of Li_4(1/2).

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A099217 Decimal expansion of Li_3(1/2).

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A099219 Decimal expansion of Li_5(1/2).

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A099220 Decimal expansion of Li_6(1/2).

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A099221 Decimal expansion of Li_7(1/2).

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A099222 Decimal expansion of Li_8(1/2).

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A099223 Decimal expansion of Li_9(1/2).

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