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%I #3 Mar 30 2012 18:36:58
%S 1,1,3,6,17,42,114,302,827,2263,6275,17468,48967,137834,389738,
%T 1105861,3148240,8987989,25726635,73808069,212196040,611219900,
%U 1763659860,5097131364,14752847173,42757853357,124080269331,360493591232
%N G.f. satisfies: A(x) = G(x)*A(x^2*G(x)) where G(x) is the g.f. of the Motzkin numbers (A001006): G = (1 + x*G + x^2*G^2).
%C Equals column 0 of triangle A121400.
%e A(x) = 1 + x + 3*x^2 + 6*x^3 + 17*x^4 + 42*x^5 + 114*x^6 +...
%e The g.f. of the Motzkin numbers begins:
%e G(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 51*x^6 + 127*x^7 +...
%o (PARI) {a(n)=local(F=1+x+x^2,G=serreverse(x/(F+x^2*O(x^n)))/x,H=1+x,A); for(i=0,n,H=G*subst(H,x,x^2*G)+x^2*O(x^n)); A=(x*H-y*subst(H,x,x*y))/(x*subst(F,x,y)-y); polcoeff(polcoeff(A,n,x),0,y)}
%Y Cf. A121400 (triangle), A121398 (main diagonal), A001006 (Motzkin).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 27 2006
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