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%I #15 Oct 08 2018 21:09:31
%S 1,1,2,3,5,8,12,20,28,45,65,101,148,221,316,469,673,969,1420,2025,
%T 2892,4100,5905,8314,11860,16645,23399,32838,46071,64274,89761,124977,
%U 173231,240492,332978,460015,634271,874464,1200463,1649499,2263102,3098661,4239109
%N G.f.: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + n*x^d/d) ).
%C Note: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + x^d) ) does not yield an integer series.
%F Logarithmic derivative yields A205477.
%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 12*x^6 + 20*x^7 + ...
%e By definition:
%e log(A(x)) = x*(1+x) + x^2*(1+2*x)*(1+x^2)/2 + x^3*(1+3*x)*(1+x^3)/3 + x^4*(1+4*x)*(1+2*x^2)*(1+x^4)/4 + x^5*(1+5*x)*(1+x^5)/5 + x^6*(1+6*x)*(1+3*x^2)*(1+2*x^3)*(1+x^6)/6 + ...
%e Explicitly,
%e log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 11*x^5/5 + 12*x^6/6 + 29*x^7/7 + 15*x^8/8 + 49*x^9/9 + 43*x^10/10 + ... + A205477(n)*x^n/n + ...
%t max = 50; s = Exp[Sum[(x^n/n)*Product[1+n*x^d/d, {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+m*x^d/d+x*O(x^n)))))), n)}
%Y Cf. A205477 (log), A198304, A205478, A205480, A205482, A205484, A205486, A205488, A205490.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 27 2012
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