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%I #4 Jan 21 2018 16:17:43
%S 1,1,3,11,47,221,1117,5981,33619,197139,1200551,7567125,49233845,
%T 329945065,2273469967,16082532495,116649264071,866551528737,
%U 6586844135753,51188050930421,406394722000439,3294052336807639,27243245715300079,229773018019419769,1975311828734850201
%N G.f. A(x) satisfies: A(x - x^2 - x^2*A(x)) = x.
%C Compare to: C(x - x^2) = x where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%H Paul D. Hanna, <a href="/A295539/b295539.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) also satisfies:
%F (1) A(x) = x + A(x)^2 * (A(A(x)) + 1).
%F (2) A(x) = Series_Reversion(x - x^2 - x^2*A(x)).
%F (3) A(A(x)) = (A(x) - x)/A(x)^2 - 1.
%e G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 47*x^5 + 221*x^6 + 1117*x^7 + 5981*x^8 + 33619*x^9 + 197139*x^10 + 1200551*x^11 + 7567125*x^12 + 49233845*x^13 + 329945065*x^14 + 2273469967*x^15 + 16082532495*x^16 + ...
%e such that A(x - x^2 - x^2*A(x)) = x.
%e RELATED SERIES.
%e A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 28*x^5 + 125*x^6 + 602*x^7 + 3079*x^8 + 16556*x^9 + 92973*x^10 + 542750*x^11 + 3282515*x^12 + 20513732*x^13 + 132193781*x^14 + 876924910*x^15 + 5979574323*x^16 + ...
%e A(A(x)) = x + 2*x^2 + 8*x^3 + 38*x^4 + 202*x^5 + 1156*x^6 + 6990*x^7 + 44158*x^8 + 289344*x^9 + 1956846*x^10 + 13612042*x^11 + 97142544*x^12 + 709885514*x^13 + 5304302214*x^14 + 40479776540*x^15 + 315231061286*x^16 + ...
%e which equals (A(x) - x)/A(x)^2 - 1.
%o (PARI) {a(n) = my(A=x); for(i=1,n\2, A = serreverse(x-x^2 - x^2*A +x*O(x^n))); polcoeff(A,n)}
%o for(n=1,30,print1(a(n),", "))
%o (PARI) {a(n) = my(A=x); for(i=1,n, A = x + A^2 * subst(1+A,x,A +x*O(x^n))); polcoeff(A,n)}
%o for(n=1,30,print1(a(n),", "))
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jan 21 2018
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