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%I #5 Mar 23 2022 17:29:50
%S 1,1,2,5,12,30,78,206,552,1498,4105,11340,31541,88237,248076,700478,
%T 1985397,5646129,16104378,46056513,132031176,379315946,1091890772,
%U 3148736064,9095091878,26310816944,76219704957,221085782559,642058752476,1866693825362,5432795508417
%N G.f. satisfies: A(x) = G(x)*A(x^3*G(x)^2) where G(x) is the g.f. of the Motzkin numbers (A001006).
%C Equals column 0 and main diagonal of triangle A120894 (cascadence of 1+x+x^2).
%e A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 30*x^5 + 78*x^6 + 206*x^7+...
%e = G(x)*A(x^3*G(x)^2) where
%e G(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 51*x^6 + 127*x^7 +...
%e is the g.f. of the Motzkin numbers (A001006) so that G(x) satisfies:
%e G(x) = 1 + x*G(x) + x^2*G(x)^2.
%o (PARI) {a(n)=local(A=1+x,G=1/x*serreverse(x/(1+x+x^2+x*O(x^n)))); for(i=0,n,A=G*subst(A,x,x^3*G^2 +x*O(x^n)));polcoeff(A,n,x)}
%Y Cf. A120894, A120896, A120897; A001006; variants: A092684, A092687, A120899.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 14 2006
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