A251690 G.f. A(x) satisfies the condition that G(A(x)) is a power series in x consisting entirely of positive integer coefficients such that G(A(x) - x^k) has negative coefficients for k>0, where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

1, -1, -2, -2, 0, -1, 0, -3, 0, -3, -3, 0, -3, -2, -3, -1, -2, 0, -1, -2, 0, 0, -2, 0, 0, 0, -2, 0, -3, 0, -2, 0, -1, 0, -3, -2, -1, -1, -3, -1, 0, -2, -2, -3, -1, -3, -1, -1, 0, 0, -1, -1, -3, -3, -1, 0, -1, 0, -2, 0, -3, -3, -3, -2, -1, -2, -1, -2, -2, -2, -3, -1, -3, -1, -3, -1, 0, -2, -2, -2, -1, -1, -2, -2, 0, -3, -3, -2, -3, -1, -3, -2, 0, 0, 0, -2, -2, -2, -2, -3, -3, 0, -2, 0, -3, -1, 0, -2, -3, -1, -3, 0, -1, 0, -2, -1, -1, -3, -3, -1, -3, -3, 0, -3, -2, -3, -2

Offset

1,3

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).

It seems that a(n) lies in [-3,0] for n>1; does this continue to hold?

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Example

G.f.: A(x) = x - x^2 - 2*x^3 - 2*x^4 - x^6 - 3*x^8 - 3*x^10 - 3*x^11 - 3*x^13 - 2*x^14 - 3*x^15 - x^16 - 2*x^17 - x^19 - 2*x^20 - 2*x^23 - 2*x^27 - 3*x^29 +...
Given G(x) = 1 + x*G(x)^3, which begins
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 +...
then
G(A(x)) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 17*x^5 + 36*x^6 + 78*x^7 + 169*x^8 + 370*x^9 + 813*x^10 + 1793*x^11 + 3971*x^12 +...+ A251691(n)*x^n +...
consists entirely of positive integer coefficients such that
G(A(x) - x^k) has negative coefficients for k>0, as illustrated by:
k=1: G(A(x) - x) = 1 - x^2 - 2*x^3 + x^4 + 12*x^5 + 11*x^6 - 48*x^7 +...
k=2: G(A(x) - x^2) = 1 + x + x^2 - 2*x^3 - 19*x^4 - 83*x^5 - 267*x^6 +...
k=3: G(A(x) - x^3) = 1 + x + 2*x^2 + 3*x^3 + 2*x^4 - 13*x^5 - 97*x^6 - 471*x^7 +...
k=4: G(A(x) - x^4) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 11*x^5 + 6*x^6 - 58*x^7 - 413*x^8 +...
k=5: G(A(x) - x^5) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 16*x^5 + 30*x^6 + 48*x^7 + 33*x^8 - 215*x^9 - 1632*x^10 +...
k=6: G(A(x) - x^6) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 17*x^5 + 35*x^6 + 72*x^7 + 139*x^8 + 234*x^9 + 228*x^10 - 655*x^11 - 6102*x^12 +...
etc.
The position of the first negative term in G(A(x) - x^n), n>=1, begins:
[2, 3, 5, 7, 9, 11, 13, 16, 18, 20, 23, 25, 28, 30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 55, 57, 59, 62, 64, 67, 69, 72, 74, 77, 79, 82, 84, 87, 89, 92, 94, 97, 99, 102, 104, 106, 109, 111, 114, 116, 119, ...];
does the ratio of the position of the first negative term in G(A(x)-x^n) divided by n approach some constant?

Prog

(PARI) /* Prints initial N terms: */
N=100;
/* G(x) = 1 + x*G(x)^3 is the g.f. of A001764: */
{G=1+serreverse(x/(1+x +x*O(x^(3*N+10)))^3); }
/* Print terms as you build vector A, then print A at the end: */
{A=[1, -1]; print1("1, -1, ");
for(l=1, N, A=concat(A, -4);
for(i=1, 4, A[#A]=A[#A]+1;
V=Vec(subst(G, x, x*truncate(Ser(A)) +O(x^floor(3*#A+1)) ));
if((sign(V[3*#A])+1)/2==1, print1(A[#A], ", "); break)); ); A}

Crossrefs

Cf. A251691, A001764.

Sequence in context: A263571 A364060 A361292 * A187752 A295181 A215573

Adjacent sequences: A251687 A251688 A251689 * A251691 A251692 A251693

Keyword

sign

Author

Paul D. Hanna, Dec 31 2014

Status

approved