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%I #3 Mar 30 2012 18:36:58
%S 1,3,12,55,276,1464,8058,45543,262626,1538607,9130446,54761628,
%T 331403447,2021021082,12407102937,76611488305,475493441604,
%U 2964664310319,18560063203353,116621922800283,735236268006654
%N G.f. satisfies: A(x) = G(x)^3 * A(x^4*G(x)^9), where G(x) is the g.f. of the number of ternary trees (A001764): G(x) = 1 + x*G(x)^3.
%C Column 0 of triangle A120919 (cascadence of (1+x)^3).
%e A(x) = 1 + 3*x + 12*x^2 + 55*x^3 + 276*x^4 + 1464*x^5 + 8058*x^6 +...
%e = G(x)^3 * A(x^4*G(x)^9) where
%e G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
%e is g.f. of A001764: G(x) = 1 + x*G(x)^3.
%o (PARI) {a(n)=local(A=1+x,G=(1/x*serreverse(x/(1+3*x+3*x^2+x^3+x*O(x^n))))^(1/3)); for(i=0,n,A=G^3*subst(A,x,x^4*G^9 +x*O(x^n)));polcoeff(A,n,x)}
%Y Cf. A120919, A120921 (cube-root), A120922, A120923; A001764; variants: A092684, A092687, A120895, A120899, A120920.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 17 2006
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