%I #5 Jun 16 2014 11:31:59
%S 3,7,2,8,9,7,1,4,3,2,8,6,7,2,8,9,9,4,2,2,0,2,1,1,2,2,8,7,6,2,1,1,4,6,
%T 0,2,1,7,6,3,5,9,2,9,2,0,0,0,4,6,7,3,7,5,7,9,5,7,8,4,9,1,7,6,7,2,4,8,
%U 9,4,6,2,1,5,3,8,5,0,7,7,7,9,6,3,0,6,7,5,7,3,9,8,0,1,0,4,5,7,6,2,9
%N Decimal expansion of the variance of the maximum of a size 8 sample from a normal (0,1) distribution.
%C According to Steven Finch, no exact expression of this moment is known.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.
%F integral_(-infinity..infinity) 8*x^2*F(x)^7*f(x) dx - mu(8)^2, where f(x) is the normal (0,1) density and F(x) its cumulative distribution, mu(8) being the moment A243961.
%e 0.3728971432867289942202112287621146...
%t digits = 101; m0 = 5; dm = 5; f[x_] := 1/ Sqrt[2*Pi]*Exp[-x^2/2]; F[x_] := 1/2*Erf[x/Sqrt[2]] + 1/2; Clear[mu8]; mu8[m_] := mu8[m] = 8*NIntegrate[x*F[x]^7*f[x], {x, -m , m}, WorkingPrecision -> digits+5, MaxRecursion -> 20]; mu8[m0]; mu8[m = m0+dm]; While[RealDigits[mu8[m]] != RealDigits[mu8[m-dm]], Print["m1 = ", m]; m = m+dm]; m8 = mu8[m]; Clear[v, m]; v[m_] := v[m] = 8*NIntegrate[x^2*F[x]^7*f[x], {x, -m , m}, WorkingPrecision -> digits+5, MaxRecursion -> 20]; v[m0]; v[m = m0+dm]; While[RealDigits[v[m]] != RealDigits[v[m-dm]], Print["m2 = ", m]; m = m+dm]; v8 = v[m]-m8^2; RealDigits[v8, 10, digits] // First
%Y Cf. A188340 v(2), A243447 v(3), A243452 v(4), A243454 v(5), A243525 v(6), A243526 v(7), A243961 mu(8).
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Jun 16 2014