A258814 - OEIS

%I #16 Nov 07 2023 04:56:42

%S 9,9,9,5,5,4,5,0,7,8,9,0,5,3,9,9,0,9,4,9,6,3,4,6,5,4,9,8,9,9,0,5,8,9,

%T 8,3,0,0,2,1,8,8,4,8,1,9,4,9,9,7,5,7,9,2,2,5,2,6,4,9,2,1,8,9,4,1,9,0,

%U 1,1,2,1,4,4,5,9,1,1,0,5,0,0,0,6,7,5,7,8,6,6,7,9,9,5,3,6,6,4,2,0,8,8

%N Decimal expansion of the Dirichlet beta function of 7.

%H G. C. Greubel, <a href="/A258814/b258814.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletBetaFunction.html">Dirichlet Beta Function</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_beta_function">Dirichlet beta function</a>.

%F beta(7) = Sum_{n>=0} (-1)^n/(2n+1)^7 = (zeta(7, 1/4) - zeta(7, 3/4))/16384 = 61*Pi^7/184320.

%F Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^7)^(-1). - _Amiram Eldar_, Nov 06 2023

%e 0.9995545078905399094963465498990589830021884819499757922526492189419...

%t RealDigits[DirichletBeta[7], 10, 102] // First

%o (PARI) 61*Pi^7/184320 \\ _Charles R Greathouse IV_, Dec 06 2016

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); 61*Pi(R)^7/184320; // _G. C. Greubel_, Aug 24 2018

%Y Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A258815 (beta(8)), A258816 (beta(9)).

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Jun 11 2015