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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, 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
0,3
EXAMPLE
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 +...
PROG
(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)))}
CROSSREFS
Sequence in context: A001521 A003143 A221718 * A017983 A139077 A017825
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 19 2015
STATUS
approved