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A289090 Decimal expansion of (E(|x|^3))^(1/3), with x being a normally distributed random variable.

%I A289090 #15 Jul 28 2017 13:36:28
%S A289090 1,1,6,8,5,7,5,2,5,4,9,6,2,4,6,5,5,4,8,6,7,0,4,7,6,0,1,1,0,9,7,6,8,5,
%T A289090 2,7,1,0,6,0,5,2,4,0,4,8,1,6,7,9,0,7,9,7,2,3,8,3,5,1,6,2,8,7,4,2,3,4,
%U A289090 1,5,2,9,3,8,8,8,7,8,5,4,6,5,2,7,8,7,1,4,2,3,4,2,8,3,8,3,4,9,3,9,6,7,3,1,3
%N A289090 Decimal expansion of (E(|x|^3))^(1/3), with x being a normally distributed random variable.
%C A289090 The p-th root r(p) of the expected value E(|x|^p) for various distributions appears, for example, in chemical physics, where some interactions depend on high powers of interatomic distances.
%C A289090 When x is distributed normally with zero mean and standard deviation 1, r(p) evaluates to r(p) = ((p-1)!!*w(p))^(1/p), where w(p) = 1 for even p and sqrt(2/Pi) for odd p. Note that, by definition, r(2) = 1 and r(1) = w(1) = A076668.
%C A289090 The present constant is a = r(3).
%H A289090 Stanislav Sykora, <a href="/A289090/b289090.txt">Table of n, a(n) for n = 1..1000</a>
%H A289090 Wikipedia, <a href="https://en.wikipedia.org/wiki/Normal_distribution">Normal distribution</a>
%F A289090 Equals (2!!*sqrt(2/Pi))^(1/3) = (2*A076668)^(1/3).
%e A289090 1.16857525496246554867047601109768527106052404816790797238351628742...
%t A289090 ExpectedValue[Abs[#]^3&, NormalDistribution[0, 1]]^(1/3) // RealDigits[#, 10, 105]& // First (* _Jean-François Alcover_, Jul 28 2017 *)
%o A289090 (PARI) \\ General code, for any p > 0:
%o A289090 r(p) = (sqrt(2/Pi)^(p%2)*prod(k=0,(p-2)\2,p-1-2*k))^(1/p);
%o A289090 a = r(3) \\ Present instance
%Y A289090 Cf. A060294, A076668 (p=1), A011002 (p=4), A289091 (p=5), A011350 (p=6).
%K A289090 nonn,cons
%O A289090 1,3
%A A289090 _Stanislav Sykora_, Jul 26 2017