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%I #15 Aug 20 2016 22:57:13
%S 1,1,2,6,19,65,234,873,3346,13099,52154,210541,859768,3545263,
%T 14741148,61736903,260192880,1102704585,4696416190,20090502706,
%U 86285786519,371917832707,1608317086940,6975728777332,30338392601498,132277349730004,578075052215714,2531710609461484,11109852467209553,48843541287179595,215108137824940916,948874606956945665,4191979050580762418,18545890698661636784,82159569800859439840,364432560308538162214,1618431087549954575022
%N G.f. satisfies: A(x) = x + A( A(x)^2 - A(x)^5 ).
%C Compare to: C(x) = x + C( C(x)^2 - C(x)^4 ) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%H Paul D. Hanna, <a href="/A275755/b275755.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. satisfies:
%F (1) A(x - A(x^2 - x^5)) = x.
%F (2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^2-x^5)^(n+1) / (n+1)!.
%F (3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^2-x^5)^(n+1)/x / (n+1)! ).
%e G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 19*x^5 + 65*x^6 + 234*x^7 + 873*x^8 + 3346*x^9 + 13099*x^10 + 52154*x^11 + 210541*x^12 + 859768*x^13 + 3545263*x^14 +...
%e such that A(x) = x + A( A(x)^2 - A(x)^5 ).
%e RELATED SERIES.
%e A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 54*x^6 + 192*x^7 + 710*x^8 + 2702*x^9 + 10515*x^10 + 41660*x^11 + 167483*x^12 + 681532*x^13 + 2801816*x^14 +...
%e A(x)^5 = x^5 + 5*x^6 + 20*x^7 + 80*x^8 + 320*x^9 + 1286*x^10 + 5210*x^11 + 21285*x^12 + 87655*x^13 + 363660*x^14 + 1518952*x^15 +...
%e A(x^2 - x^5) = x^2 + x^4 - x^5 + 2*x^6 - 2*x^7 + 6*x^8 - 6*x^9 + 20*x^10 - 24*x^11 + 71*x^12 - 95*x^13 + 270*x^14 - 392*x^15 + 1063*x^16 - 1662*x^17 +...
%e where Series_Reversion(A(x)) = x - A(x^2 - x^5).
%o (PARI) {a(n) = my(A=x); for(i=1,n, A = x + subst(A,x, A^2 - A^5 +x*O(x^n))); polcoeff(A,n)}
%o for(n=1,40,print1(a(n),", "))
%Y Cf. A275756, A275757, A275758, A141200.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Aug 20 2016
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