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A210453 Decimal expansion of Sum_{n>=1} 1/(n*binomial(3*n,n)).

%I A210453 #26 Dec 07 2024 05:43:28
%S A210453 3,7,1,2,1,6,9,7,5,2,6,0,2,4,7,0,3,4,4,7,4,7,7,1,6,6,6,0,7,5,3,5,8,8,
%T A210453 0,5,5,8,7,6,2,9,4,6,9,0,5,1,9,7,2,2,2,1,3,6,4,7,7,8,9,3,9,5,7,3,4,0,
%U A210453 0,0,8,3,5,3,5,5,9,8,4,9,6,9,1,3,1,4,3,2,7,5,4,1,7,7,6,5,0,5,0,9,9,2,3,2,3,9,6,1,7,5,6,9,0,7,7,3,5,3,5,2,7,3,1,6,8,6
%N A210453 Decimal expansion of Sum_{n>=1} 1/(n*binomial(3*n,n)).
%D A210453 George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press, 2006, p. 60.
%H A210453 Necdet Batir, <a href="https://doi.org/10.1007/BF02829799">On the series Sum_{k=1..oo} binomial(3k,k)^{-1} k^{-n} x^k</a>, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 4 (2005), pp. 371-381; <a href="https://arxiv.org/abs/math/0512310">arXiv preprint</a>, arXiv:math/0512310 [math.AC], 2005.
%H A210453 Courtney Moen, <a href="http://www.jstor.org/stable/2690456">Infinite series with binomial coefficients</a>, Math. Mag. 64 (1) (1991) 53-55.
%F A210453 Equals Sum_{n>=1} 1/(n*A005809(n)).
%F A210453 Equals Integral_{x=0..1} x^2/(1-x^2+x^3) dx.
%F A210453 Equals Sum_(R) R*log(1-1/R)/(3*R-2) where R is summed over the set of the three constants -A075778, A210462-i*A210463 and A210462-i*A210463, i=sqrt(-1), that is, over the set of the three roots of x^3-x^2+1.
%F A210453 Equals (1/sqrt(23)) * (arctan(sqrt(3)/(2*phi-1)) * 18*phi/(phi^2-phi+1) - log((phi^3+1)/(phi+1)^3) * (3*sqrt(3)*phi*(1-phi))/(phi^3+1)), where phi = ((25+3*sqrt(69))/2)^(1/3) (Batir, 2005, p. 378, eq. (3.2)). - _Amiram Eldar_, Dec 07 2024
%e A210453 0.37121697526024703447477166607535880558762946905197...
%p A210453 A075778neg := proc()
%p A210453         1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;
%p A210453 end proc:
%p A210453 A210462 := proc()
%p A210453         local a075778 ;
%p A210453         a075778 := A075778neg() ;
%p A210453         (1+1/a075778/(a075778-1))/2 ;
%p A210453 end proc:
%p A210453 A210463 := proc()
%p A210453         local a075778,a210462 ;
%p A210453         a075778 := A075778neg() ;
%p A210453         a210462 := A210462() ;
%p A210453         -1/a075778-a210462^2 ;
%p A210453         sqrt(%) ;
%p A210453 end proc:
%p A210453 A210453 := proc()
%p A210453         local v,x;
%p A210453         v := 0.0 ;
%p A210453         for x in [ A075778neg(), A210462()+I*A210463(), A210462()-I*A210463() ] do
%p A210453                 v := v+ x*log(1-1/x)/(3*x-2) ;
%p A210453         end do:
%p A210453         evalf(v) ;
%p A210453 end proc:
%p A210453 A210453() ;
%t A210453 RealDigits[ HypergeometricPFQ[{1, 1, 3/2}, {4/3, 5/3}, 4/27]/3, 10, 105] // First (* _Jean-François Alcover_, Feb 11 2013 *)
%Y A210453 Cf. A005809, A073010, A075778, A210462, A210463.
%K A210453 cons,nonn
%O A210453 0,1
%A A210453 _R. J. Mathar_, Jan 21 2013