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%I #24 Jan 15 2018 09:54:39
%S 1,1,2,6,18,59,198,690,2450,8878,32632,121518,457262,1736526,6646340,
%T 25613086,99298674,387021728,1515594560,5960406102,23530528512,
%U 93216984177,370450977206,1476458287082,5900150928510,23635544130948
%N G.f. satisfies A(x) = 1 + x*A(x)^2*A(x^2).
%H Vaclav Kotesovec, <a href="/A036675/b036675.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1-x^2/(1-x^2)) (continued fraction); more generally g.f. C(x/(1-x^2/(1-x^2))) where C(x) is the g.f. for the Catalan numbers (A000108). [_Joerg Arndt_, Mar 18 2011]
%F a(n) ~ c * d^n / n^(3/2), where d = 4.250770453055989899189676464071962617426..., c = 0.600960911911396921862654605015399962... . - _Vaclav Kotesovec_, Aug 10 2014
%F a(n) = T(2*n+1,1), where T(n,m) = sum(i=1..n-m, (m*binomial(m+2*i-1,i))/(m+i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i)), n>m, T(n,n)=1. - _Vladimir Kruchinin_, Mar 18 2015
%p A := 1; f := proc(n) global A; coeff(series( 1+x*(A*subs(x=x^2,A)), x, n+1), x,n); end; for n from 1 to 50 do A := series(A+f(n)*x^n,x,n +1); od: A;
%t terms = 26; A[_] = 0; Do[A[x_] = 1 + x*A[x]^2*A[x^2] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* _Jean-François Alcover_, Jan 15 2018 *)
%o (PARI) a(n)=local(A,m); if(n<0,0,m=2; A=1+O(x); while(m<=n+1,m*=2; A=2/(1+sqrt(1-4*x*subst(A,x,x^2)))); polcoeff(A,n))
%o (Maxima)
%o T(n,m):=if m=n then 1 else sum((m*binomial(m+2*i-1,i))/(m+i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i),i,1,n-m);
%o makelist(T(2*n+1,1),n,0,30); /* _Vladimir Kruchinin_, Mar 18 2015 */
%Y Cf. A000621, A101913
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_.
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