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A205480 G.f: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^(n/d))^d ).

%I A205480 #13 Oct 08 2018 21:08:28
%S A205480 1,1,2,4,10,27,76,242,852,3016,11262,47004,204761,894673,4134909,
%T A205480 20370101,101904474,521459464,2813783214,15616060213,87143803196,
%U A205480 502477538546,3039137586808,18763942581733,116737580008529,742909490860950,4846956807516551
%N A205480 G.f: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^(n/d))^d ).
%C A205480 Note: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + x^(n/d))^d ) does not yield an integer series.
%F A205480 Logarithmic derivative yields A205481.
%e A205480 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 27*x^5 + 76*x^6 + 242*x^7 + ...
%e A205480 By definition:
%e A205480 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^2)^2*(1+4*x)^4/4 + x^5*(1+x^5)*(1+5*x)^5/5 + x^6*(1+x^6)*(1+2*x^3)^2*(1+3*x^2)^3*(1+6*x)^6/6 + ...
%e A205480 Explicitly,
%e A205480 log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 76*x^5/5 + 249*x^6/6 + 974*x^7/7 + 4151*x^8/8 + 16558*x^9/9 + ... + A205481(n)*x^n/n + ...
%t A205480 max = 30; s = Exp[Sum[(x^n/n)*Product[(1+d*x^(n/d))^d, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s , x] (* _Jean-François Alcover_, Dec 23 2015 *)
%o A205480 (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, d*log(1+d*x^(m/d)+x*O(x^n)))))), n)}
%Y A205480 Cf. A205481 (log), A205476, A205478, A205482, A205484, A205486, A205488, A205490.
%K A205480 nonn
%O A205480 0,3
%A A205480 _Paul D. Hanna_, Jan 27 2012