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%I #19 Apr 15 2016 09:22:20
%S 1,1,1,2,6,17,45,123,356,1061,3193,9691,29741,92228,288331,907316,
%T 2871818,9138352,29217870,93813839,302367222,977913351,3172727729,
%U 10323213290,33677758165,110135406578,360982631871,1185632194394,3901702645568,12862978506084,42477650804971,140496123308659,465383254510707,1543693890316144
%N G.f. A(x) satisfies: A( A(x)^2 - A(x)^3 ) = x*A(x).
%C Compare g.f. to: C( C(x)^2 - C(x)^4 ) = C(x)^2 and C( C(x) - C(x)^2 ) = C(x) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%H Paul D. Hanna, <a href="/A268655/b268655.txt">Table of n, a(n) for n = 1..1030</a>
%F G.f. A(x) satisfies: A( A(x^2-x^3)/x ) = x.
%F a(n) ~ c * d^n / n^(3/2), where d = 3.469666051910765293806107674644687565... and c = 0.1297182904218030477707168121291985... . - _Vaclav Kotesovec_, Apr 01 2016
%e G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 17*x^6 + 45*x^7 + 123*x^8 + 356*x^9 + 1061*x^10 + 3193*x^11 + 9691*x^12 + 29741*x^13 + 92228*x^14 + 288331*x^15 +...
%e such that A( A(x)^2 - A(x)^3 ) = x*A(x).
%e RELATED SERIES.
%e A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 17*x^6 + 50*x^7 + 140*x^8 + 394*x^9 + 1152*x^10 + 3464*x^11 + 10541*x^12 + 32320*x^13 + 99973*x^14 +...
%e A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 36*x^7 + 108*x^8 + 316*x^9 + 918*x^10 + 2727*x^11 + 8291*x^12 + 25521*x^13 + 79110*x^14 + 246942*x^15 +...
%e A(x)^2 - A(x)^3 = x^2 + x^3 + 4*x^6 + 14*x^7 + 32*x^8 + 78*x^9 + 234*x^10 + 737*x^11 + 2250*x^12 + 6799*x^13 + 20863*x^14 + 65056*x^15 +...
%e The series reversion of g.f. A(x) equals A(x^2-x^3)/x, which begins:
%e A(x^2-x^3)/x = x - x^2 + x^3 - 2*x^4 + 2*x^5 - 3*x^6 + 5*x^7 - 9*x^8 + 18*x^9 - 38*x^10 + 79*x^11 - 162*x^12 + 330*x^13 - 661*x^14 + 1323*x^15 - 2661*x^16 + 5392*x^17 - 11037*x^18 + 22802*x^19 - 47447*x^20 + 99238*x^21 - 208283*x^22 +...
%o (PARI) /* From definition A( A(x)^2 - A(x)^3 ) = x*A(x) */
%o {a(n) = my(A=[1,1]); for(i=1,n, A = concat(A,0); G=x*Ser(A); A[#A] = -Vec(subst(G,x,G^2 - G^3 ))[#A] ); A[n]}
%o for(n=1,40,print1(a(n),", "))
%o (PARI) /* Using Series Reversion of A(x^2-x^3)/x */
%o {a(n) = my(A=x); for(i=1,n, A = serreverse( subst(A,x, x^2 - x^3 +x^2*O(x^n))/x) ); polcoeff(A,n)}
%o for(n=1,40,print1(a(n),", "))
%K nonn
%O 1,4
%A _Paul D. Hanna_, Apr 01 2016
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