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

0,3

Links

Table of n, a(n) for n=0..47.

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

Cf. A251570, A251691.

Sequence in context: A001521 A003143 A221718 * A017983 A139077 A017825

Adjacent sequences: A251568 A251569 A251570 * A251572 A251573 A251574

Keyword

nonn

Author

Paul D. Hanna, Jan 19 2015

Status

approved