cp's OEIS Frontend

A205488 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n)^d ).

%I A205488 #8 Aug 22 2025 17:36:05
%S A205488 1,1,2,3,7,9,26,32,85,129,293,389,1105,1426,3242,5121,11405,15790,
%T A205488 37704,52289,118470,181770,373257,543137,1250814,1796474,3713201,
%U A205488 5790866,11919993,17663189,37445802,55499537,113653306,177038255,347919239,529686395,1091858128
%N A205488 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n)^d ).
%C A205488 Note: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - x^n)^d ) does not yield an integer series.
%F A205488  Logarithmic derivative yields A205489.
%e A205488 G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 9*x^5 + 26*x^6 + 32*x^7 +...
%e A205488 By definition:
%e A205488 log(A(x)) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x^2)^2) + (x^3/3)/((1-x^3)*(1-3*x^3)^3) + (x^4/4)/((1-x^4)*(1-2*x^4)^2*(1-4*x^4)^4) + (x^5/5)/((1-x^5)*(1-5*x^5)^5) + (x^6/6)/((1-x^6)*(1-2*x^6)^2*(1-3*x^6)^3*(1-6*x^6)^6) +...
%e A205488 Explicitly,
%e A205488 log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 6*x^5/5 + 78*x^6/6 + 8*x^7/7 + 247*x^8/8 + 202*x^9/9 + 708*x^10/10 +...+ A205489(n)*x^n/n +...
%o A205488 (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -d*log(1-d*x^m+x*O(x^n)))))), n)}
%Y A205488 Cf. A205489 (log), A205476, A205478, A205480, A205482, A205484, A205486, A205490.
%K A205488 nonn,changed
%O A205488 0,3
%A A205488 _Paul D. Hanna_, Jan 27 2012