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%I #5 Jun 06 2014 08:22:21
%S 3,9,1,9,1,7,7,7,6,1,2,6,7,5,0,4,5,2,8,1,9,6,8,4,9,6,5,8,0,0,0,9,1,9,
%T 9,8,7,2,0,2,2,0,9,9,1,2,2,1,1,3,0,8,1,8,7,4,1,9,6,8,0,7,0,6,3,7,4,5,
%U 8,7,3,4,6,1,9,3,3,5,8,6,8,4,4,3,5,8,2,5,1,4,1,6,5,2,8,8,2,6,3
%N Decimal expansion of the variance of the maximum of a size 7 sample from a normal (0,1) distribution.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.
%F -((441*(Pi*(Pi - 5*arccsc(sqrt(3))) + 5*integral_(x=0..arccsc(sqrt(3)) )(arcsin(sqrt(3)*sqrt(1/(8 - tan(x)^2))), {x, 0, arccsc(sqrt(3))} ))^2)/(4*Pi^5)) + (35*sqrt(3)*(Pi*(Pi - 4*arccsc(2*sqrt(2/3))) + 2*integral_(x=0..arccsc(2*sqrt(2/3)))(arcsin(sqrt(6)*sqrt(1/(15 - tan(x)^2))), {x, 0, arccsc(2*sqrt(2/3))} )))/(4*Pi^3) + 1
%e 0.39191777612675045281968496580009199872...
%t digits = 99; v[7] = -((441*(Pi*(Pi - 5*ArcCsc[Sqrt[3]]) + 5*NIntegrate[ArcSin[Sqrt[3]*Sqrt[1/(8 - Tan[x]^2)]], {x, 0, ArcCsc[Sqrt[3]]},
%t WorkingPrecision -> digits + 5])^2)/(4*Pi^5)) + (35*Sqrt[3]*(Pi*(Pi - 4*ArcCsc[2*Sqrt[2/3]]) + 2*NIntegrate[ArcSin[Sqrt[6]*Sqrt[1/(15 - Tan[x]^2)]], {x, 0, ArcCsc[2*Sqrt[2/3]]}, WorkingPrecision -> digits + 5]))/(4*Pi^3) + 1; RealDigits[v[7], 10, digits] // First
%Y Cf. A188340, A243447, A243452, A243454, A243525, A243524.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Jun 06 2014