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%I #20 Nov 01 2019 18:36:33
%S 1,1,1,2,4,12,35,133,497,2256,10123,53131,276210,1638039,9639943,
%T 63526677,416194299,3009639922,21672348693,170290649517,1334332599748,
%U 11302630861664,95587196023618,867197921850406,7862652321850812,75983785567389333,734442008292947615
%N G.f. satisfies: A(x) = 1 + x*A(x/A(-x)).
%H Robert Israel, <a href="/A211768/b211768.txt">Table of n, a(n) for n = 0..220</a>
%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 12*x^5 + 35*x^6 + 133*x^7 +...
%e Related expansion:
%e x/A(-x) = x + x^2 + x^4 - x^5 + 6*x^6 - 14*x^7 + 72*x^8 - 250*x^9 + 1338*x^10 +..
%p eq:= 1 - A(x) + x*A(x/A(-x)):
%p AA[0]:= 1: c[0]:= 1:
%p for n from 1 to 50 do
%p S:= series(eval(eq, A = unapply(AA[n-1]+c[n]*x^n, x)), x, n+1);
%p c[n]:= solve(convert(S,polynom),c[n]);
%p AA[n]:= AA[n-1]+c[n]*x^n;
%p od:
%p seq(c[n],n=0..50); # _Robert Israel_, Aug 02 2017
%t nmax = 26; sol = {a[0] -> 1};
%t Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x*A[x/A[-x]]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t sol /. HoldPattern[a[n_] -> k_] :> Set[a[n], k];
%t a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 01 2019 *)
%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x*subst(A,x,x/subst(A,x,-x+x*O(x^n)))); polcoeff(A, n)}
%o for(n=0,25,print1(a(n),", "))
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jun 06 2012
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