A251570 - OEIS

%I #7 Jan 06 2015 00:44:36

%S 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,

%T -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,

%U -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

%N 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.

%C 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).

%H Paul D. Hanna, <a href="/A251570/b251570.txt">Table of n, a(n) for n = 1..300</a>

%e 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 +...

%e Given the g.f. M(x) 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 +...

%e then

%e 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 +...

%e consists entirely of positive integer coefficients such that M(A(x) - x^k) has negative coefficients for k>0.

%e Note that a(n) = -2 seems somewhat sparse and occurs at positions:

%e [58, 123, 181, 187, 203, 213, 230, 236, 245, 253, ...].

%o (PARI) /* Prints initial N 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 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)); ); A}

%Y Cf. A251571, A251690, A001006.

%K sign

%O 1,58

%A _Paul D. Hanna_, Jan 06 2015