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%I #7 Jun 24 2014 17:39:03
%S 1,1,3,8,25,87,323,1213,4616,18122,73034,296118,1201985,4944712,
%T 20699593,87039567,364582540,1536503788,6577636822,28301509234,
%U 120717294622,515067346615,2238588931727,9820016463003,42372035446400,181068476007679,795531553724263,3577424216585403
%N G.f. satisfies: A(x*A(-x)) = C(x), where C(x) = 1 + x*C(x)^2 is the Catalan function of A000108.
%F G.f. satisfies: A(x) = 1 + A(x)^2 * Series_Reversion(x*A(-x)).
%e G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 87*x^5 + 323*x^6 + 1213*x^7 +...
%e where
%e A(x*A(-x)) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 1430*x^8 +...+ A000108(n)*x^n +...
%e Related expansions:
%e A(x)^2 = 1 + 2*x + 7*x^2 + 22*x^3 + 75*x^4 + 272*x^5 + 1034*x^6 +...
%e Series_Reversion(x*A(-x)) = x + x^2 - x^3 - 2*x^4 + x^5 + 10*x^6 - x^7 - 80*x^8 - 35*x^9 + 841*x^10 + 956*x^11 +...
%o (PARI) /* From formula: A(x*A(-x)) = C(x) = 1 + x*C(x)^2 */
%o {a(n)=local(A=1+x,C=sum(m=0,n,binomial(2*m,m)/(m+1)*x^m));
%o for(i=1,n,A=subst(C,x,serreverse(x*subst(A,x,-x) +x*O(x^n))));polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* A(x) = 1 + A(x)^2*Series_Reversion(x*A(-x)): */
%o {a(n)=local(A=1+x);for(i=1,n,A=1+A^2*serreverse(x*subst(A,x,-x) +x*O(x^n)));polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A000108.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 24 2014
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