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 A257084 G.f. A(x) satisfies A(F(x)) = x, where F(x) is the g.f. of A251690. 0
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 13 07:00 EDT 2024. Contains 375865 sequences. (Running on oeis4.)