%I #13 Sep 17 2023 21:26:03
%S 1,1,3,7,22,53,189,485,1688,4689,16650,47187,165262,485268,1728805,
%T 5174302,18049100,55269365,194672136,603386084,2099060047,6614696429,
%U 23137388920,73539215187,254877900451,819954155438,2849069767691,9230855030167,31891296634949,104243641176924
%N G.f. satisfies: A(x) = x*exp( Sum_{n>=1} A(sigma(n)*x^n) / n ).
%C Compare to: G(x) = x*exp( Sum_{n>=1} G(x^n)/n ), which is the g.f. of A000081, the number of rooted trees with n nodes.
%C Compare to: exp( Sum_{n>=1} sigma(n)*x^n/n ) = Product_{n>=1} 1/(1-x^n), which is the g.f. of the partition numbers (A000041).
%e G.f.: A(x) = x + x^2 + 3*x^3 + 7*x^4 + 22*x^5 + 53*x^6 + 189*x^7 + 485*x^8 + ...
%e where
%e A(x) = x*exp(A(x) + A(3*x^2)/2 + A(4*x^3)/3 + A(7*x^4)/4 + A(6*x^5)/5 + A(12*x^6)/6 + A(8*x^7)/7 + A(15*x^8)/8 + A(13*x^9)/9 + A(18*x^10)/10 + ...).
%o (PARI) {a(n)=local(A=x);for(i=1,n,A=x*exp(sum(k=1,n,subst(A,x,sigma(k)*x^k +x*O(x^n))/k)));polcoeff(A,n)}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A000203 (sigma), A229900, A229901.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Oct 02 2013