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A230283 Numerators to Dirichlet inverse of Euler totient based version of series expansion for x/LambertW(x).

%I A230283 #21 Sep 15 2024 14:02:57
%S A230283 1,1,-1,2,-9,8,-625,2,-117649,128,-6561,8,-25937424601,18,
%T A230283 -23298085122481,16,-9,32768,-48661191875666868481,400,
%U A230283 -104127350297911241532841,648,-81,256,-907846434775996175406740561329,490,-59604644775390625,1024,-2541865828329,1296
%N A230283 Numerators to Dirichlet inverse of Euler totient based version of series expansion for x/LambertW(x).
%C A230283 The coefficients of the series expansion of x/Lambert(x) expanded at 0 can be seen as exponentiated numerators in convergents of zeta function limits of truncated Dirichlet series for logarithms. Those numerators are defined by simple recurrences. Letting those recurrences run in cross directions to each other, one get the Dirichlet inverse of the Euler totient in a greatest common divisor matrix, and the von Mangoldt function as convergents of Dirichlet series. Since x/LambertW(x) is good at approximately describing the nontrivial Riemann zeta zeros and since the Riemann zeta zeros are the frequencies that build up the von Mangoldt function, this prime number or von Mangoldt function version of the x/LambertW(x) is motivated.
%H A230283 G. C. Greubel, <a href="/A230283/b230283.txt">Table of n, a(n) for n = 1..385</a>
%t A230283 Clear[nn, n, k, s, x]; nn = 22; Numerator[CoefficientList[1 + Integrate[1 + Expand[Sum[Exp[Limit[Zeta[s]*Sum[(If[n == 1, 0, Table[DivisorSum[m, # MoebiusMu[#] &], {m, nn}][[GCD[n, k]]]])/(k)^(s - 1), {k, 1, n}], s -> 1]]*(-x)^n, {n, 1, nn}]], x], x]]
%Y A230283 Cf. A191898, A177885, A230284 (denominators).
%K A230283 sign,frac
%O A230283 1,4
%A A230283 _Mats Granvik_, Oct 15 2013