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A205489 L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n)^d.

%I A205489 #8 Aug 22 2025 17:57:10
%S A205489 1,3,4,15,6,78,8,247,202,708,12,4146,14,5498,8964,24135,18,81114,20,
%T A205489 206520,193736,225558,24,2314378,242656,1278332,3622954,9209950,30,
%U A205489 26654118,32,58890983,59213598,35652216,28736938,628796418,38,179307278,878319368
%N A205489 L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n)^d.
%F A205489 Forms the logarithmic derivative of A205488.
%e A205489 L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 6*x^5/5 + 78*x^6/6 +...
%e A205489 By definition:
%e A205489 L(x) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x^2)^2) + (x^3/3)/((1-x^3)*(1-3*x^3)^3) + (x^4/4)/((1-x^4)*(1-2*x^4)^2*(1-4*x^4)^4) + (x^5/5)/((1-x^5)*(1-5*x^5)^5) + (x^6/6)/((1-x^6)*(1-2*x^6)^2*(1-3*x^6)^3*(1-6*x^6)^6) +...
%e A205489 Exponentiation yields the g.f. of A205488:
%e A205489 exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 9*x^5 + 26*x^6 + 32*x^7 +...
%o A205489 (PARI) {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -d*log(1-d*x^m+x*O(x^n))))), n)}
%Y A205489 Cf. A205488 (exp), A205477, A205479, A205481, A205483, A205485, A205487, A205491.
%K A205489 nonn,changed
%O A205489 1,2
%A A205489 _Paul D. Hanna_, Jan 27 2012