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%I #4 May 15 2016 06:37:23
%S 1,1,1,2,2,3,5,9,18,38,79,162,330,661,1323,2661,5392,11037,22802,
%T 47447,99238,208283,438143,923325,1949051,4121495,8731982,18536690,
%U 39428284,84023511,179370023,383518886,821198510,1760683462,3779593676,8122853103,17476215940,37639236974,81146453958,175111467257,378230792221,817669121153,1769125092131,3830738971497,8301063679980,18001035450869,39062530229674,84822294102377,184304055379313,400704466515940
%N G.f. A(x) satisfies: A(x*A(-x)) = x^3 - x^2.
%H Paul D. Hanna, <a href="/A273034/b273034.txt">Table of n, a(n) for n = 1..520</a>
%e G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 5*x^7 + 9*x^8 + 18*x^9 + 38*x^10 + 79*x^11 + 162*x^12 + 330*x^13 + 661*x^14 + 1323*x^15 + 2661*x^16 +...
%e such that A(x*A(-x)) = x^3 - x^2.
%e RELATED SERIES.
%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 + x^3 - 2*x^4 + 6*x^5 - 17*x^6 + 45*x^7 - 123*x^8 + 356*x^9 - 1061*x^10 + 3193*x^11 - 9691*x^12 + 29741*x^13 - 92228*x^14 +...+ (-1)^(n-1)*A268655(n)*x^n +...
%e where B(x^3 - x^2) = x*A(-x),
%e also, B(B(x^3-x^2)/x) = -x.
%o (PARI) {a(n) = my(A=[1,1],F); for(i=1,n, A=concat(A,0); F=x*Ser(A); A[#A] = Vec(subst(F,x,-x*F))[#A]); A[n]}
%o for(n=1,50,print1(a(n),", "))
%Y Cf. A268655.
%K nonn
%O 1,4
%A _Paul D. Hanna_, May 15 2016
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