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%I #3 Jan 19 2015 23:46:46
%S 1,1,2,3,4,6,9,13,19,27,39,55,79,113,160,228,322,455,641,902,1268,
%T 1777,2490,3483,4864,6791,9468,13189,18358,25527,35473,49248,68336,
%U 94751,131288,181815,251627,348051,481180,664885,918285,1267663,1749212,2412635,3326303,4584236,6315428,8697260
%N G.f.: M(F(x)) is a power series in x consisting entirely of positive integer coefficients such that M(F(x) - x^k) has negative coefficients for k>0, where M(x) = 1 + x*M(x) + x*M(x)^2 is the g.f. of the Motzkin numbers A001006.
%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 13*x^7 +...
%e such that A(x) = M(F(x)),
%e where F(x) is the g.f. of A251570:
%e F(x) = x - x^3 - x^4 + x^5 - x^7 - x^8 + x^10 - x^11 - x^13 - x^14 - x^16 - x^17 - x^18 - x^20 - x^22 - x^26 - x^27 - x^28 - x^29 - x^32 - x^33 - x^35 - x^36 - x^39 - x^41 - x^43 - x^44 - x^45 - x^46 - x^47 - x^48 - x^50 +...
%e and M(x) is the g.f. of the Motzkin numbers:
%e M(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 51*x^6 + 127*x^7 + 323*x^8 + 835*x^9 + 2188*x^10 + 5798*x^11 + 15511*x^12 +...
%o (PARI) /* Prints initial N+2 terms: */
%o N=100;
%o /* M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of Motzkin numbers: */
%o {M=1/x*serreverse(x/(1+x+x^2 +x*O(x^(2*N+10)))); M +O(x^21) }
%o /* Print terms as you build vector A, then print a(n) at the end: */
%o {A=[1, 0]; print1("1, 0, ");
%o for(l=1, N, A=concat(A, -3);
%o for(i=1, 4, A[#A]=A[#A]+1;
%o V=Vec(subst(M, x, x*truncate(Ser(A)) +O(x^floor(2*#A+1)) ));
%o if((sign(V[2*#A])+1)/2==1, print1(A[#A], ", "); break)); );
%o Vec(subst(M,x,x*Ser(A)))}
%Y Cf. A251570, A251691.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 19 2015
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