%I #14 Oct 08 2018 21:08:58
%S 1,1,2,2,4,4,8,8,14,15,24,25,43,45,69,74,113,120,187,198,291,314,452,
%T 483,720,770,1089,1182,1657,1784,2530,2724,3764,4102,5593,6053,8361,
%U 9049,12183,13304,17831,19378,26097,28355,37548,41107,54031,58894,78008,85052
%N G.f.: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + d*x^n) ).
%C Note: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + x^n) ) does not yield an integer series.
%F Logarithmic derivative yields A205479.
%e G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 + ...
%e By definition:
%e log(A(x)) = x*(1+x) + x^2*(1+x^2)*(1+2*x^2)/2 + x^3*(1+x^3)*(1+3*x^3)/3 + x^4*(1+x^4)*(1+2*x^4)*(1+4*x^4)/4 + x^5*(1+x^5)*(1+5*x^5)/5 + x^6*(1+x^6)*(1+2*x^6)*(1+3*x^6)*(1+6*x^6)/6 + ...
%e Explicitly,
%e log(A(x)) = x + 3*x^2/2 + x^3/3 + 7*x^4/4 + x^5/5 + 15*x^6/6 + x^7/7 + 15*x^8/8 + 10*x^9/9 + 13*x^10/10 + ... + A205479(n)*x^n/n + ...
%t max = 50; s = Exp[Sum[(x^n/n)*Product[1+d*x^n, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s, x] (* _Jean-François Alcover_, Dec 23 2015 *)
%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, log(1+d*x^m+x*O(x^n)))))), n)}
%Y Cf. A205479, A198296, A205476, A205480, A205482, A205484, A205486, A205488, A205490.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 27 2012