cp's OEIS Frontend

A205486 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^(n/d))^d ).

%I A205486 #8 Aug 22 2025 17:57:22
%S A205486 1,1,2,5,16,60,259,1273,7048,43241,289685,2097912,16317134,135574160,
%T A205486 1196898329,11168544771,109647222799,1128440311914,12139734936953,
%U A205486 136195813530558,1590028534430967,19277087785530470,242235954813757132,3149491477171141810
%N A205486 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^(n/d))^d ).
%C A205486 Note: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - x^(n/d))^d ) does not yield an integer series.
%F A205486 Logarithmic derivative yields A205487.
%e A205486 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 60*x^5 + 259*x^6 + 1273*x^7 +...
%e A205486 By definition:
%e A205486 log(A(x)) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x)^2) + (x^3/3)/((1-x^3)*(1-3*x)^3) + (x^4/4)/((1-x^4)*(1-2*x^2)^2*(1-4*x)^4) + (x^5/5)/((1-x^5)*(1-5*x)^5) + (x^6/6)/((1-x^6)*(1-2*x^3)^2*(1-3*x^2)^3*(1-6*x)^6) +...
%e A205486 Explicitly,
%e A205486 log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 43*x^4/4 + 206*x^5/5 + 1104*x^6/6 + 6581*x^7/7 + 43227*x^8/8 +...+ A205487(n)*x^n/n +...
%o A205486 (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 A205486 Cf. A205487 (log), A205476, A205478, A205480, A205482, A205484, A205488, A205490.
%K A205486 nonn
%O A205486 0,3
%A A205486 _Paul D. Hanna_, Jan 27 2012