%I #16 Jan 07 2019 03:46:39

%S 1,1,1,3,5,10,18,45,98,242,569,1360,3101,6799,14674,31297,66483,

%T 144954,326150,787455,2019266,5425240,14793678,40049832,105958990,

%U 272562864,679336804,1644951169,3882700822,8997217790,20559767303

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

%H G. C. Greubel, <a href="/A192860/b192860.txt">Table of n, a(n) for n = 0..1000</a>

%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 10*x^5 + 18*x^6 + 45*x^7 + ...

%e where the logarithm of the g.f. begins:

%e log(A(x)) = x + (x + 2*x^2)^2/2 + (x + 3*x^3)^3/3 + (x + 2*x^2 + 4*x^4)^4/4 + (x + 5*x^5)^5/5 + (x + 2*x^2 + 3*x^3 + 6*x^6)^6/6 + (x + 7*x^7)^7/7 + (x + 2*x^2 + 4*x^4 + 8*x^8)^8/8 + ...

%e Explicitly, the logarithm yields the l.g.f. of A192859, which begins:

%e log(A(x)) = x + x^2/2 + 7*x^3/3 + 9*x^4/4 + 26*x^5/5 + 37*x^6/6 + 162*x^7/7 + ...

%t With[{m = 35}, CoefficientList[Series[Exp[Sum[(Sum[d*x^d, {d, Divisors[n] }])^n/n, {n, 1, m + 2}]], {x, 0, m}], x]] (* _G. C. Greubel_, Jan 06 2019 *)

%o (PARI) {a(n)=local(A);A=exp(sum(m=1,n+1,sumdiv(m,d,d*x^d +x*O(x^n))^m/m));polcoeff(A,n)}

%Y Cf. A192859, A192890, A192891.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jul 11 2011