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

%I A257084 #11 Apr 17 2015 14:52:26
%S A257084 1,1,4,17,80,407,2160,11859,66754,383210,2234921,13204685,78870454,
%T A257084 475453371,2888991879,17675743626,108801199823,673302178725,
%U A257084 4186513098755,26142455226568,163873586066647,1030820865387599,6504789754356175,41166205256238155,261217480924768212,1661598566523216015
%N A257084 G.f. A(x) satisfies A(F(x)) = x, where F(x) is the g.f. of A251690.
%C A257084 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.
%e A257084 G.f.: A(x) = x + x^2 + 4*x^3 + 17*x^4 + 80*x^5 + 407*x^6 + 2160*x^7 +...
%e A257084 such that the series reversion of A(x) yields the g.f. F(x) of A251690:
%e A257084 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 -...
%e A257084 in which all coefficients after the first are in the interval [-3,0].
%e A257084 RELATED SERIES.
%e A257084 Given G(x) = 1 + x*G(x)^3, which begins
%e A257084 G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 +...
%e A257084 then
%e A257084 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 +...
%e A257084 consists entirely of positive integer coefficients such that G(F(x) - x^k) has negative coefficients for k>0.
%e A257084 Also, a related series is defined by the limits:
%e A257084 1/F'(x) = Limit ( A(F(x) + x^n) - x ) / x^n, and
%e A257084 1/F'(x) = Limit ( x - A(F(x) - x^n) ) / x^n, where
%e A257084 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 +...
%o A257084 (PARI) /* Prints initial N terms: */
%o A257084 N=50;
%o A257084 /* G(x) = 1 + x*G(x)^3 is the g.f. of A001764: */
%o A257084 {G=1+serreverse(x/(1+x +x*O(x^(3*N+10)))^3); }
%o A257084 /* Build the series reversion, then print coefficients at end: */
%o A257084 {A=[1, -1]; for(l=1, N, A=concat(A, -4);
%o A257084 for(i=1, 4, A[#A]=A[#A]+1;
%o A257084 V=Vec(subst(G, x, x*truncate(Ser(A)) +O(x^floor(3*#A+1)) ));
%o A257084 if((sign(V[3*#A])+1)/2==1, print1(".");break)););
%o A257084 Vec(serreverse(x*Ser(A)))}
%Y A257084 Cf. A251690, A251691, A001764.
%K A257084 nonn
%O A257084 1,3
%A A257084 _Paul D. Hanna_, Apr 15 2015