%I #16 Aug 08 2014 06:37:32

%S 1,1,1,4,24,194,1910,21906,284714,4118920,65499703,1134357186,

%T 21241909288,427614219498,9209265391295,211306412407522,

%U 5146853147702218,132652421589097114,3607267257131723404,103227487053425987122,3101214050804214019991,97597988749147752540052

%N G.f. satisfies: A(x) = A(x/A(x))^2 - x.

%C What is the limit (a(n+1)/a(n))/n ? (Value is near 1.44605... at n=300.)

%C Limit n->infinity (a(n+1)/a(n))/n = 1/log(2) = 1.4426950408889634... . - _Vaclav Kotesovec_, Aug 08 2014

%H Paul D. Hanna, <a href="/A241000/b241000.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. satisfies:

%F (1) A(x)^2/(1+x) = A( x * A(x)^2/(1+x) ).

%F (2) A(x)^2/(1+x) = G(x) where G(x) = A(x*G(x)) and A(x) = G(x/A(x)) and G(x*G(x)) = G(x)^2/(1 + x*G(x)).

%F a(n) ~ c * n^(n + 1/2 + log(2)) / (exp(n) * (log(2))^n), where c = 0.2812864532720972025... . - _Vaclav Kotesovec_, Aug 08 2014

%e G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 24*x^4 + 194*x^5 + 1910*x^6 + 21906*x^7 +...

%e Related series:

%e A(x)^2 = 1 + 2*x + 3*x^2 + 10*x^3 + 57*x^4 + 444*x^5 + 4272*x^6 + 48212*x^7 +...

%e A(x/A(x)) = 1 + x + 2*x^3 + 10*x^4 + 87*x^5 + 866*x^6 + 10067*x^7 +...

%e A(x/A(x))^2 = 1 + 2*x + x^2 + 4*x^3 + 24*x^4 + 194*x^5 + 1910*x^6 + 21906*x^7 +...

%o (PARI) {a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0);F=Ser(A);A[#A]=Vec(1+F-subst(F^2,x,x/F))[#A]);A[n+1]}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A240996, A240997, A240999.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Aug 07 2014