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%I #5 Dec 02 2014 08:28:12
%S 1,2,5,16,56,208,804,3202,13046,54120,227812,970596,4177436,18136008,
%T 79326641,349241700,1546401412,6882164584,30767887372,138114569908,
%U 622270344740,2812994557488,12755010576700,57996678526684
%N G.f. satisfies: A(x)^(1/2) = x + A(x) + A(A(x)) + A(A(A(x))) + A(A(A(A(x)))) +...
%H Vaclav Kotesovec, <a href="/A141201/b141201.txt">Table of n, a(n) for n = 2..250</a>
%F G.f.: A(x) = G(x)^2 where G(x) = g.f. of A141200.
%F a(n) ~ c / (n^(3/2) * r^n), where r = 0.2064501590536889244980412... (see A141200) and c = 0.051350746503831964896433888... . - _Vaclav Kotesovec_, Dec 02 2014
%e G.f.: A(x) = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 56*x^6 + 208*x^7 +...
%e A(x)^(1/2) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 72*x^6 + 272*x^7 +...
%e Related expansions:
%e A(A(x)) = x^4 + 4*x^5 + 16*x^6 + 64*x^7 + 260*x^8 + 1072*x^9 +...
%e A(A(A(x))) = x^8 + 8*x^9 + 48*x^10 + 256*x^11 + 1290*x^12 +...
%e A(A(A(A(x)))) = x^16 + 16*x^17 + 160*x^18 + 1280*x^19 + 8980*x^20 +...
%e where A(x)^(1/2) = x + A(x) + A(A(x)) + A(A(A(x))) + ...
%o (PARI) {a(n)=local(A=x+x^2);for(i=0,n,A=serreverse(x-subst(A,x,x^2+x^2*O(x^n))));polcoeff(A^2,n)}
%Y Cf. A141200.
%K nonn
%O 2,2
%A _Paul D. Hanna_, Jun 12 2008
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