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A262784 G.f. satisfies: A(x) = exp( Sum_{n>=1} A(x)^n * (x^n/n) / (1 + x^n) ).

%I A262784 #6 Sep 06 2024 18:40:07
%S A262784 1,1,1,3,7,18,51,147,431,1295,3954,12219,38174,120373,382559,1224173,
%T A262784 3940964,12754732,41476011,135446194,444016332,1460619970,4819995330,
%U A262784 15951789030,52932499724,176073789065,587010353666,1961118581900,6564548926877,22013613082514,73945428978360,248781470115309
%N A262784 G.f. satisfies: A(x) = exp( Sum_{n>=1} A(x)^n * (x^n/n) / (1 + x^n) ).
%C A262784 Compare to: exp( Sum_{n>=1} (x^n/n)/(1+x^n) ) = Sum_{n>=0} x^(n*(n+1)/2).
%e A262784 G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 7*x^4 + 18*x^5 + 51*x^6 + 147*x^7 +...
%e A262784 where
%e A262784 log(A(x)) = A(x)*x/(1+x) + A(x)^2*(x^2/2)/(1+x^2) + A(x)^3*(x^3/3)/(1+x^3) + A(x)^4*(x^4/4)/(1+x^4) + A(x)^5*(x^5/5)/(1+x^5) +...
%e A262784 explicitly,
%e A262784 log(A(x)) = x + x^2/2 + 7*x^3/3 + 17*x^4/4 + 56*x^5/5 + 187*x^6/6 + 617*x^7/7 + 2033*x^8/8 + 6811*x^9/9 + 22906*x^10/10 +...
%o A262784 (PARI) {a(n) = local(A=1+x); for(i=1,n, A = exp( sum(k=1,n, A^k*x^k/k/(1+x^k +x*O(x^n)))));polcoeff(A,n)}
%o A262784 for(n=0,40,print1(a(n),", "))
%Y A262784 Cf. A145268.
%K A262784 nonn
%O A262784 0,4
%A A262784 _Paul D. Hanna_, Oct 01 2015