A251570 G.f. A(x) satisfies the condition that M(A(x)) is a power series in x consisting entirely of positive integer coefficients such that M(A(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, 0, -1, -1, 1, 0, -1, -1, 0, 1, -1, 0, -1, -1, 0, -1, -1, -1, 0, -1, 0, -1, 0, 0, 0, -1, -1, -1, -1, 0, 0, -1, -1, 0, -1, -1, 0, 0, -1, 0, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, -1, -1, -1, -1, 0, 0, -1, 0, -1, 0, -1, 0, -1, -1, -1, 0, -1, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, -1, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -2, 0, 0, 0, 0, 0, 0, -1

Offset: 1

Author

Paul D. Hanna, Jan 06 2015

Keywords

• sign

Comments

Compare to the similar series F(x) for the Catalan function C(x) = 1 + x*C(x)^2, where C(F(x)) consists entirely of positive integer coefficients such that C(F(x) - x^k) has negative coefficients for k>0; in which case F(x) = (x+x^2) - (x+x^2)^2, and C(F(x)) = 1/(1-x-x^2).

Examples

			G.f.: A(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 +...
Given the g.f. M(x) 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 +...
then
M(A(x)) = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 13*x^7 + 19*x^8 + 27*x^9 + 39*x^10 + 55*x^11 + 79*x^12 + 113*x^13 + 160*x^14 +...+ A251571(n)*x^n +...
consists entirely of positive integer coefficients such that M(A(x) - x^k) has negative coefficients for k>0.
Note that a(n) = -2 seems somewhat sparse and occurs at positions:
[58, 123, 181, 187, 203, 213, 230, 236, 245, 253, ...].
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A251571, A251690, A001006.

Programs

PARI

/* Prints initial N 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 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)); ); A}