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%I #3 Mar 30 2012 18:36:58
%S 1,4,20,116,720,4656,30996,210896,1459536,10239796,72651184,520328112,
%T 3756512912,27307671040,199705789248,1468209751856,10844681408064,
%U 80437588353600,598867568439828,4473784063109904,33524058847464912
%N G.f. satisfies: A(x) = C(2x)^2 * A(x^3*C(2x)^4), where C(x) is the g.f. of the Catalan numbers (A000108).
%C Column 0 of triangle A120914 (cascadence of (1+2x)^2).
%e A(x) = 1 + 4*x + 20*x^2 + 116*x^3 + 720*x^4 + 4656*x^5 + 30996*x^6 +...
%e = C(2x)^2 * A(x^3*C(2x)^4) where
%e C(2x) = 1 + 2*x + 8*x^2 + 40*x^3 + 224*x^4 + 1344*x^5 + 8448*x^6 +...
%e and C(x) is 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+4*x+4*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. A120914, A120916 (square-root), A120917, A120918; A000108; variants: A092684, A092687, A120895, A120899, A120920.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 17 2006
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