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A138739 G.f. A(x) satisfies: A(A(x)) = 3*A(x) - 2*x - x^2 with A(0)=0.

%I A138739 #5 Sep 24 2017 00:09:43
%S A138739 1,1,2,11,88,888,10572,143214,2159154,35702442,640873656,12394383780,
%T A138739 256762580460,5671209169168,133041670286160,3304034094162183,
%U A138739 86616702087692256,2390831825522972392,69323685702986714272,2107073248164657741448,67003070810599639419680,2225053954972969636237280,77034579373254666948386880,2776183496539544726567249520
%N A138739 G.f. A(x) satisfies: A(A(x)) = 3*A(x) - 2*x - x^2 with A(0)=0.
%C A138739 All self-compositions of A(x) may be expressed as a finite sum involving powers of A(x) and x.
%H A138739 Paul D. Hanna, <a href="/A138739/b138739.txt">Table of n, a(n) for n = 1..300</a>
%e A138739 G.f.: A(x) = x + x^2 + 2*x^3 + 11*x^4 + 88*x^5 + 888*x^6 + 10572*x^7 + 143214*x^8 + 2159154*x^9 + 35702442*x^10 + 640873656*x^11 + 12394383780*x^12 + 256762580460*x^13 + 5671209169168*x^14 + 133041670286160*x^15 +...
%e A138739 A(A(x)) = x + 2*x^2 + 6*x^3 + 33*x^4 + 264*x^5 + 2664*x^6 + 31716*x^7 + 429642*x^8 + 6477462*x^9 + 107107326*x^10 + 1922620968*x^11 + 37183151340*x^12 + 770287741380*x^13 + 17013627507504*x^14 + 399125010858480*x^15 +...
%e A138739 so that A(A(x)) + 2*x + x^2 = 3*A(x).
%e A138739 Self-compositions of A=A(x) may be expressed in terms of A and x:
%e A138739 A(A(x)) = 3*A - 2*x - x^2 ;
%e A138739 A(A(A(x))) = (7*A - A^2) - 6*x - 3*x^2 ;
%e A138739 A(A(A(A(x)))) = (15*A - 12*A^2) + (-14 + 12*A)*x +
%e A138739 (-11 + 6*A)*x^2 - 4*x^3 - x^4 ;
%e A138739 A(A(A(A(A(x))))) = (31*A - 83*A^2 + 14*A^3 - A^4) +
%e A138739 (-12*A^2 + 120*A - 30)*x + (-6*A^2 + 60*A - 63)*x^2 - 48*x^3 - 12*x^4 .
%o A138739 (PARI) {a(n)=local(A=x+x^2);if(n<1,0, for(i=3,n+1,A=A+polcoeff(subst(A,x,A+x*O(x^i)),i)*x^i);polcoeff(A,n))}
%o A138739 for(n=1,20,print1(a(n),", "))
%Y A138739 Cf. A138740.
%K A138739 nonn
%O A138739 1,3
%A A138739 _Paul D. Hanna_, Mar 27 2008