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A370439 Expansion of g.f. A(x) satisfying A(x) = A( x*A(x)^2 + 3*x*A(x)^3 )^(1/3).

%I A370439 #16 May 26 2024 16:18:03
%S A370439 1,3,9,30,126,648,3591,19953,110079,610500,3440493,19742616,114918138,
%T A370439 675417474,3996992547,23791052862,142393544757,856746349992,
%U A370439 5179722791274,31449875426622,191678795532801,1172198278949454,7190652243631437,44235165115911312,272837082264574914
%N A370439 Expansion of g.f. A(x) satisfying A(x) = A( x*A(x)^2 + 3*x*A(x)^3 )^(1/3).
%C A370439 Compare the g.f. to the following identities:
%C A370439 (1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
%C A370439 (2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
%C A370439 where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%H A370439 Paul D. Hanna, <a href="/A370439/b370439.txt">Table of n, a(n) for n = 1..500</a>
%F A370439 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370439 (1.a) A(x)^3 = A( x*A(x)^2 * (1 + 3*A(x)) ).
%F A370439 (1.b) A(x)^9 = A( x*A(x)^8 * (1 + 3*A(x))*(1 + 3*A(x)^3) ).
%F A370439 (1.c) A(x)^27 = A( x*A(x)^26 * (1 + 3*A(x))*(1 + 3*A(x)^3)*(1 + 3*A(x)^9) ).
%F A370439 (1.d) A(x)^(3^n) = A( x*A(x)^(3^n-1) * Product_{k=0..n-1} (1 + 3*A(x)^(3^k)) ).
%F A370439 (2) A(x) = x * Product_{n>=0} (1 + 3*A(x)^(3^n)).
%F A370439 (3) A(x) = Series_Reversion( x / Product_{n>=0} (1 + 3*x^(3^n)) ).
%F A370439 (4) A(x) = x * Sum_{n>=0} A117940(n) * A(x)^n, where g.f. of A117940 equals Product{k>=0} 1 + 3*x^(3^k).
%F A370439 a(n) ~ c * d^n / n^(3/2), where d = 6.5583689184153129045048... and c = 0.129061736750222730297... - _Vaclav Kotesovec_, Apr 05 2024
%F A370439 The radius of convergence r of g.f. A(x) and A(r) satisfy 1 = Sum_{n>=0} 3^(n+1) * A(r)^(3^n) / (1 + 3*A(r)^(3^n)) and r = A(r) / Product_{n>=0} (1 + 3*A(r)^(3^n)), where r = 0.1524769363297159918479... = 1/d (d is given above) and A(r) = 0.3905308673397427979651361312666180120359942797557... - _Paul D. Hanna_, May 22 2024
%e A370439 G.f.: A(x) = x + 3*x^2 + 9*x^3 + 30*x^4 + 126*x^5 + 648*x^6 + 3591*x^7 + 19953*x^8 + 110079*x^9 + 610500*x^10 + 3440493*x^11 + 19742616*x^12 + ...
%e A370439 where A(x)^3 = A( x*A(x)^2 + 3*x*A(x)^3 ).
%o A370439 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A=concat(A,0);
%o A370439 F=Ser(A); A[#A] = polcoeff(subst(F,x,x*F^2 + 3*x*F^3) - F^3,#A+1) );A[n+1]}
%o A370439 for(n=1,30, print1(a(n),", "))
%Y A370439 Cf. A356782, A117940, A370546.
%K A370439 nonn
%O A370439 1,2
%A A370439 _Paul D. Hanna_, Mar 27 2024