%I #3 Mar 30 2012 18:36:58
%S 1,2,5,16,54,186,654,2338,8463,30938,114022,423096,1579049,5922512,
%T 22309350,84354388,320020227,1217689680,4645693038,17766596202,
%U 68092473570,261486788434,1005962436536,3876412305114,14960183283203
%N G.f. satisfies: A(x) = C(x)^2 * A(x^3*C(x)^4), where C(x) is the g.f. of the Catalan numbers (A000108).
%C Column 0 of triangle A120898 (cascadence of 1+2x+x^2). Self-convolution of A120900.
%e A(x) = 1 + 2*x + 5*x^2 + 16*x^3 + 54*x^4 + 186*x^5 + 654*x^6 +...
%e = C(x)^2 * A(x^3*C(x)^4) where
%e C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
%e is the g.f. of the Catalan numbers (A000108): C(x) = 1 + x*C(x)^2.
%o (PARI) {a(n)=local(A=1+x,C=(1/x*serreverse(x/(1+2*x+x^2+x*O(x^n))))^(1/2)); for(i=0,n,A=C^2*subst(A,x,x^3*C^4 +x*O(x^n)));polcoeff(A,n,x)}
%Y Cf. A120898, A120900 (square-root), A120901, A120902; A000108; variants: A092684, A092687, A120895.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 14 2006
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