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A274181 Decimal expansion of Phi(1/2, 2, 2), where Phi is the Lerch transcendent. 6
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=0..100.

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.

FORMULA

c = Phi(1/2, 2, 2) with Phi the Lerch transcendent.

c = Sum_{k>=0}(1/((2+k)^2*2^k)).

c = 4 * polylog(2, 1/2) - 2.

c = Pi^2/3 - 2*log(2)^2 - 2.

EXAMPLE

0.32896210586005002361062528063872043497679389922...

MAPLE

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

MATHEMATICA

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

PROG

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

(Python)

from mpmath import *

mp.dps=102

print map(int, list(str(lerchphi(1/2, 2, 2))[2:-1])) # Indranil Ghosh, Jul 04 2017

CROSSREFS

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

Sequence in context: A230432 A195305 A021308 * A195055 A214683 A060921

Adjacent sequences:  A274178 A274179 A274180 * A274182 A274183 A274184

KEYWORD

cons,nonn

AUTHOR

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

STATUS

approved

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Last modified February 19 22:04 EST 2018. Contains 299357 sequences. (Running on oeis4.)