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%I #9 Dec 10 2016 03:03:15
%S 0,1,3,7,99,9307,3462205,401327263,5290639975663,21886143096656843,
%T 32306573547837099089161,2837034062676862693613762377,
%U 182184397885888753164448171682621
%N Numerators of the Engel partial sums for L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.
%D Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292
%F chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
%F Series: L(3, chi3) = Sum_{k>=1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
%F Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
%e L(3, chi3) = 0.8840238117500798567430579168710118077... = 1/2 + 1/(2*2) + 1/(2*2*2) + 1/(2*2*2*14) + 1/(2*2*2*14*94) + ..., the partial sums of which are 0, 1/2, 3/4, 7/8, 99/112, 9307/10528, ...
%t nmax = 100; prec = 2000 (* Adjust the precision depending on nmax. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; e = First@Transpose@NestList[{Ceiling[1/(#[[1]] #[[2]] - 1)], #[[1]] #[[2]] - 1}&, {Ceiling[1/c], c}, nmax - 1]; Numerator[ FoldList[Plus, 0, 1/Drop[ FoldList[Times, 1, e], 1 ] ] ]
%Y Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.
%Y Cf. A129658, A129659, A129661, A129662, A129663, A129664, A129665.
%K nonn,frac,easy
%O 0,3
%A _Stuart Clary_, Apr 30 2007
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