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%I #7 May 19 2016 18:38:11
%S 1,1,2,7,26,98,389,1617,6884,29818,131284,585966,2644084,12041541,
%T 55280357,255556439,1188632292,5558332388,26116960640,123243816854,
%U 583830832350,2775426824820,13235989547850,63306397853250,303597924249840,1459546439694004,7032721684219584,33958054240107363,164290025822878210,796288552943558821,3866056901078138147,18799970286124092609,91557579962243322708
%N G.f. A(x) satisfies: A(x)^2 - 2*A(x)^3 = -A(-x^2).
%F G.f. A(x) satisfies: -A(-B(x)^2) = x^2 - 2*x^3, where A(B(x)) = x.
%e G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 98*x^6 + 389*x^7 + 1617*x^8 + 6884*x^9 + 29818*x^10 + 131284*x^11 + 585966*x^12 +...
%e where A(x)^2 - 2*A(x)^3 = -A(-x^2).
%e RELATED SERIES.
%e A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 18*x^5 + 70*x^6 + 276*x^7 + 1127*x^8 + 4768*x^9 + 20606*x^10 + 90414*x^11 + 402210*x^12 + 1810476*x^13 +...
%e A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 34*x^6 + 138*x^7 + 567*x^8 + 2384*x^9 + 10290*x^10 + 45207*x^11 + 201154*x^12 + 905238*x^13 +...
%e where
%e A(x)^2 - 2*A(x)^3 = x^2 - x^4 + 2*x^6 - 7*x^8 + 26*x^10 - 98*x^12 + 389*x^14 - 1617*x^16 + 6884*x^18 - 29818*x^20 + 131284*x^22 - 585966*x^24 +...
%e Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then
%e B(x) = x - x^2 - 2*x^4 - x^7 - 9*x^8 + 15*x^9 - 19*x^10 - 21*x^11 + 9*x^12 - 30*x^13 - 74*x^14 - 119*x^15 + 5*x^16 - 216*x^17 - 2164*x^18 + 3937*x^19 - 10603*x^20 + 9568*x^21 - 26632*x^22 + 22777*x^23 - 63015*x^24 - 42449*x^25 + 69029*x^26 +...
%e where A(-B(x)^2) = 2*x^3 - x^2,
%e also, B(2*x^3 - x^2) = -B(x)^2.
%o (PARI) /* From -A(-B(x)^2) = x^2 - 2*x^3, where A(B(x)) = x: */
%o {a(n) = my(A=[1, 1], F, B); for(i=1, n, A=concat(A, 0); F=x*Ser(A); B=serreverse(F); A[#A] = Vec(subst(-F, x, -B^2))[#A]/2); A[n]}
%o for(n=1, 40, print1(a(n), ", "))
%Y Cf. A273095.
%K nonn
%O 1,3
%A _Paul D. Hanna_, May 19 2016
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