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G.f. A(x) satisfies: A(x) = x + A( x*A(x) + x*A(x)^3 ).
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%I #8 Apr 16 2016 04:24:23

%S 1,1,1,3,8,20,55,159,464,1383,4200,12910,40112,125832,397888,1266848,

%T 4058263,13070453,42297553,137467673,448499679,1468388784,4822816903,

%U 15886282268,52468807845,173718343364,576466929104,1916968549390,6387086400663,21319636605919,71284279000874,238724756808108,800659887614429,2689098091847122

%N G.f. A(x) satisfies: A(x) = x + A( x*A(x) + x*A(x)^3 ).

%C Compare g.f. to: C(x) = x + C( x*C(x) + x*C(x)^2 ) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

%H Paul D. Hanna, <a href="/A271843/b271843.txt">Table of n, a(n) for n = 1..300</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 3.51361449558530219727... and c = 0.1466906366440109... . - _Vaclav Kotesovec_, Apr 16 2016

%e G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 8*x^5 + 20*x^6 + 55*x^7 + 159*x^8 + 464*x^9 + 1383*x^10 + 4200*x^11 + 12910*x^12 +...

%e where A(x) = x + A( x*A(x) + x*A(x)^3 ).

%e RELATED SERIES.

%e A(x) + A(x)^3 = x + x^2 + 2*x^3 + 6*x^4 + 14*x^5 + 36*x^6 + 103*x^7 + 297*x^8 + 867*x^9 + 2598*x^10 + 7908*x^11 + 24337*x^12 + 75725*x^13 + 237822*x^14

%e +...

%o (PARI) {a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1,n, A = x + subst(A,x,x*A + x*A^3) ) ; polcoeff(A,n)}

%o for(n=1,40,print1(a(n),", "))

%K nonn

%O 1,4

%A _Paul D. Hanna_, Apr 15 2016