A251571 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.
1, 1, 2, 3, 4, 6, 9, 13, 19, 27, 39, 55, 79, 113, 160, 228, 322, 455, 641, 902, 1268, 1777, 2490, 3483, 4864, 6791, 9468, 13189, 18358, 25527, 35473, 49248, 68336, 94751, 131288, 181815, 251627, 348051, 481180, 664885, 918285, 1267663, 1749212, 2412635, 3326303, 4584236, 6315428, 8697260
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 13*x^7 +... such that A(x) = M(F(x)), where F(x) is the g.f. of A251570: 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 +... and M(x) is the g.f. of the Motzkin numbers: 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 +...
Programs
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PARI
/* Prints initial N+2 terms: */ N=100; /* M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of Motzkin numbers: */ {M=1/x*serreverse(x/(1+x+x^2 +x*O(x^(2*N+10)))); M +O(x^21) } /* Print terms as you build vector A, then print a(n) at the end: */ {A=[1, 0]; print1("1, 0, "); for(l=1, N, A=concat(A, -3); for(i=1, 4, A[#A]=A[#A]+1; V=Vec(subst(M, x, x*truncate(Ser(A)) +O(x^floor(2*#A+1)) )); if((sign(V[2*#A])+1)/2==1, print1(A[#A], ", "); break)); ); Vec(subst(M,x,x*Ser(A)))}