%I #13 Oct 08 2018 21:06:40

%S 1,1,2,3,7,14,30,65,132,280,632,1439,3299,7569,17450,40313,92889,

%T 212801,483590,1092649,2467078,5581232,12690828,29123728,67648617,

%U 159370347,381080620,923803158,2264970530,5599185887,13909201590,34612152762,86049014990

%N G.f.: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^d)^n ).

%C Note: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + x^d)^n ) does not yield an integer series.

%F Logarithmic derivative yields A205485.

%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 14*x^5 + 30*x^6 + 65*x^7 + ...

%e By definition:

%e log(A(x)) = x*(1+x) + x^2*(1+x)^2*(1+2*x^2)^2/2 + x^3*(1+x)^3*(1+3*x^3)^3/3 + x^4*(1+x)^4*(1+2*x^2)^4*(1+4*x^4)^4/4 + x^5*(1+x)^5*(1+5*x^5)^5/5 + x^6*(1+x)^6*(1+2*x^2)^6*(1+3*x^3)^6*(1+6*x^6)^6/6 + ...

%e Explicitly,

%e log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 31*x^5/5 + 72*x^6/6 + 176*x^7/7 + 327*x^8/8 + 751*x^9/9 + ... + A205485(n)*x^n/n + ...

%t max = 40; s = Exp[Sum[(x^n/n)*Product[(1 + d*x^d)^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, m*log(1+d*x^d+x*O(x^n)))))), n)}

%Y Cf. A205485 (log), A205476, A205478, A205480, A205482, A205486, A205488, A205490.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 27 2012