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A219359 G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^(n*(n+1)/2).

%I A219359 #7 Aug 28 2025 17:24:24
%S A219359 1,1,3,15,99,785,7151,73097,827501,10295549,140100323,2077648871,
%T A219359 33466994639,583535328293,10971899911703,221571516903233,
%U A219359 4786126571515961,110142864494603313,2690257065904106571,69500954207702580927,1893141166438848208747,54217997416999007587201
%N A219359 G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^(n*(n+1)/2).
%F A219359 G.f. satisfies: A(x) = B(x*A(x)) and A(x/B(x)) = B(x) where B(x) satisfies:
%F A219359 B(x) = Sum_{n>=0} n!*x^n * B(x)^(n*(n-1)/2) and is the g.f. of A219358.
%e A219359 G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 99*x^4 + 785*x^5 + 7151*x^6 +...
%e A219359 where
%e A219359 A(x) = 1 + 1!*x*A(x) + 2!*x^2*A(x)^3 + 3!*x^3*A(x)^6 + 4!*x^4*A(x)^10 + 5!*x^5*A(x)^15 +...
%o A219359 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k!*x^k*(A+x*O(x^n))^(k*(k+1)/2)));polcoeff(A, n)}
%o A219359 for(n=0, 25, print1(a(n), ", "))
%Y A219359 Cf. A155805, A107591, A219358.
%K A219359 nonn,changed
%O A219359 0,3
%A A219359 _Paul D. Hanna_, Nov 18 2012