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%I #19 Aug 28 2017 07:44:12
%S 1,1,2,4,12,36,112,361,1186,3974,13524,46612,162384,570880,2022800,
%T 7216480,25900036,93449752,338772408,1233326352,4507204720,
%U 16528765376,60805491392,224335046602,829851744732,3077246265612,11436732740472,42593968518536,158941264247584,594169284671232,2224933015422432,8344687554060528,31343475208937024,117893400330845424,444019302263216224
%N G.f. satisfies: A(x - A(x) + A(x)^2) = x^4.
%C At what positions n is a(n) odd?
%C Compare g.f. to: C(x - C(x) + C(x)^2) = 0, trivial when C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).
%H Paul D. Hanna, <a href="/A291190/b291190.txt">Table of n, a(n) for n = 1..512</a>
%F G.f. A(x) satisfies: x - A(x) + A(x)^2 = Ai(x^4) where Ai( A(x) ) = x.
%F a(n) ~ c * d^n / n^(3/2), where d = 3.93460560538976027645396919840971895891402... and c = 0.137506207625998211308202134... - _Vaclav Kotesovec_, Aug 28 2017
%e G.f.: A(x) = x + x^2 + 2*x^3 + 4*x^4 + 12*x^5 + 36*x^6 + 112*x^7 + 361*x^8 + 1186*x^9 + 3974*x^10 + 13524*x^11 + 46612*x^12 + 162384*x^13 + 570880*x^14 + 2022800*x^15 + 7216480*x^16 + 25900036*x^17 + 93449752*x^18 + 338772408*x^19 + 1233326352*x^20 + 4507204720*x^21 +...
%e where A(x - A(x) + A(x)^2) = x^4.
%e RELATED SERIES.
%e Define Ai(x) such that Ai(A(x)) = x, where Ai(x) begins:
%e Ai(x) = x - x^2 + x^4 - 4*x^5 + 6*x^6 - 28*x^8 + 92*x^9 - 146*x^10 - 36*x^11 + 968*x^12 - 3076*x^13 + 4628*x^14 + 3112*x^15 - 39947*x^16 + 119776*x^17 - 163020*x^18 - 205356*x^19 + 1800122*x^20 - 5042852*x^21 + 5978324*x^22 + 12502776*x^23 - 85355762*x^24 + 222312900*x^25 +...
%e then x - A(x) + A(x)^2 = Ai(x^4),
%e and Ai(x) - Ai( Ai(x)^4 ) = x - x^2.
%o (PARI) {a(n) = my(A=x,V=[1, 1, 2,4]); for(i=1,n, V=concat(V,0); A=x*Ser(V); V[#V]=Vec(subst(A,x,x - A + A^2))[#V-3]);V[n]}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A291189.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Aug 20 2017
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