This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A274181 #63 Feb 16 2025 08:33:36
%S A274181 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,
%T A274181 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,
%U A274181 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
%N A274181 Decimal expansion of Phi(1/2, 2, 2), where Phi is the Lerch transcendent.
%C A274181 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).
%C A274181 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.
%D A274181 William Feller, An introduction to probability theory and its applications, Vol. 1. p. 285, 1968.
%H A274181 J. W. Meijer and N. H. G. Baken, <a href="http://dx.doi.org/10.1016/0167-7152(87)90041-1">The Exponential Integral Distribution</a>, Statistics and Probability Letters, Volume 5, No. 3, April 1987. pp 209-211.
%H A274181 Eric W. Weisstein’s World of Mathematics, <a href="https://mathworld.wolfram.com/LerchTranscendent.html">Lerch transcendent</a>.
%H A274181 Eric W. Weisstein’s World of Mathematics, <a href="https://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%F A274181 Equals Phi(1/2, 2, 2) with Phi the Lerch transcendent.
%F A274181 Equals Sum_{k>=0}(1/((2+k)^2*2^k)).
%F A274181 Equals 4 * polylog(2, 1/2) - 2.
%F A274181 Equals Pi^2/3 - 2*log(2)^2 - 2.
%F A274181 Equals Integral_{x=0..oo} x*exp(-x)/(exp(x)-1/2) dx. - _Amiram Eldar_, Aug 24 2020
%e A274181 0.32896210586005002361062528063872043497679389922...
%p A274181 Digits := 101; c := evalf(LerchPhi(1/2, 2, 2));
%t A274181 N[HurwitzLerchPhi[1/2, 2, 2], 25] (* _G. C. Greubel_, Jun 19 2016 *)
%o A274181 (PARI) Pi^2/3 - 2*log(2)^2 - 2 \\ _Altug Alkan_, Jul 08 2016
%o A274181 (PARI) lerchphi(.5,2,2) \\ _Charles R Greathouse IV_, Jan 30 2025
%o A274181 (Python)
%o A274181 from mpmath import mp, lerchphi
%o A274181 mp.dps=102
%o A274181 print([int(d) for d in list(str(lerchphi(1/2, 2, 2))[2:-1])]) # _Indranil Ghosh_, Jul 04 2017
%Y A274181 Cf. A163931, A002162 (Phi(1/2, 1, 1)/2), A076788 (Phi(1/2, 2, 1)/2), A112302, A008276.
%K A274181 cons,nonn
%O A274181 0,1
%A A274181 _Johannes W. Meijer_ and N. H. G. Baken, Jun 17 2016, Jul 08 2016