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A120018 The third self-composition of A120010; g.f.: A(x) = G(G(G(x))), where G(x) = g.f. of A120010.

%I A120018 #15 Mar 11 2015 01:27:29
%S A120018 1,3,9,30,114,480,2157,10092,48525,238143,1187952,6006171,30710553,
%T A120018 158535975,825143145,4325320191,22814398392,120999555588,644878190175,
%U A120018 3451975941243,18550877091063,100047282676491,541314936448764
%N A120018 The third self-composition of A120010; g.f.: A(x) = G(G(G(x))), where G(x) = g.f. of A120010.
%C A120018 Row 3 of A120019, the square table of self-compositions of A120010.
%H A120018 Vincenzo Librandi, <a href="/A120018/b120018.txt">Table of n, a(n) for n = 1..300</a>
%F A120018 G.f.: A(x) = (1 - sqrt(1 - 4*x*(1-x)/(1-3*x+3*x^2) ))/2.
%F A120018 Recurrence: n*a(n) = 2*(5*n-6)*a(n-1) - (31*n-66)*a(n-2) + 42*(n-3)*a(n-3) - 21*(n-4)*a(n-4). - _Vaclav Kotesovec_, Oct 24 2012
%F A120018 a(n) ~ sqrt(14*sqrt(21)-42)*((7+sqrt(21))/2)^n/(16*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2012
%e A120018 A(x) = x + 3*x^2 + 9*x^3 + 30*x^4 + 114*x^5 + 480*x^6 + 2157*x^7 +...
%e A120018 G(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 18*x^6 + 53*x^7 + 158*x^8 +...
%e A120018 where G(x) is the g.f. of A120010 and G(G(G(x))) = A(x).
%t A120018 CoefficientList[Series[(1 - Sqrt[1 - 4 x (1-x) / (1 -3 x + 3 x^2)]) / x / 2,  {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 24 2012 *)
%o A120018 (PARI) {a(n)=polcoeff((1 - sqrt(1 - 4*x*(1-x)/(1-3*x+3*x^2+x*O(x^n)) ))/2, n)}
%Y A120018 Cf. A120010, A120017 (2nd self-composition), A120019.
%K A120018 nonn
%O A120018 1,2
%A A120018 _Paul D. Hanna_, Jun 14 2006
%E A120018 Typo in Mma program fixed by _Vincenzo Librandi_, May 22 2013