A257084 G.f. A(x) satisfies A(F(x)) = x, where F(x) is the g.f. of A251690.

1, 1, 4, 17, 80, 407, 2160, 11859, 66754, 383210, 2234921, 13204685, 78870454, 475453371, 2888991879, 17675743626, 108801199823, 673302178725, 4186513098755, 26142455226568, 163873586066647, 1030820865387599, 6504789754356175, 41166205256238155, 261217480924768212, 1661598566523216015

Offset

1,3

Comments

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

Links

Table of n, a(n) for n=1..26.

Example

G.f.: A(x) = x + x^2 + 4*x^3 + 17*x^4 + 80*x^5 + 407*x^6 + 2160*x^7 +...
such that the series reversion of A(x) yields the g.f. F(x) of A251690:
F(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 - 2*x^31 - x^33 - 3*x^35 - 2*x^36 - x^37 - x^38 - 3*x^39 - x^40 - 2*x^42 - 2*x^43 - 3*x^44 - x^45 - 3*x^46 - x^47 - x^48 - x^51 -...
in which all coefficients after the first are in the interval [-3,0].
RELATED SERIES.
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(F(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(F(x) - x^k) has negative coefficients for k>0.
Also, a related series is defined by the limits:
1/F'(x) = Limit ( A(F(x) + x^n) - x ) / x^n, and
1/F'(x) = Limit ( x - A(F(x) - x^n) ) / x^n, where
1/F'(x) = 1 + 2*x + 10*x^2 + 40*x^3 + 156*x^4 + 638*x^5 + 2544*x^6 + 10248*x^7 + 41152*x^8 + 165350*x^9 + 664477*x^10 + 2669644*x^11 + 10727319*x^12 + 43102392*x^13 + 173188681*x^14 + 695884096*x^15 + 2796104790*x^16 +...

Prog

(PARI) /* Prints initial N terms: */
N=50;
/* 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); }
/* Build the series reversion, then print coefficients at end: */
{A=[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("."); break)); );
Vec(serreverse(x*Ser(A)))}

Crossrefs

Cf. A251690, A251691, A001764.

Sequence in context: A206228 A089165 A056096 * A371915 A245377 A351150

Adjacent sequences: A257081 A257082 A257083 * A257085 A257086 A257087

Keyword

nonn

Author

Paul D. Hanna, Apr 15 2015

Status

approved