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
Keywords
Examples
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 +...
Programs
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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)))}
Comments