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%I #13 Nov 10 2018 06:06:03
%S 1,1,1,2,4,8,17,38,87,203,482,1160,2822,6929,17149,42736,107144,
%T 270060,683940,1739511,4441255,11378814,29245927,75386341,194838673,
%U 504802508,1310843123,3411070838,8893590454,23230151872,60780378423,159281034709
%N Series reversion of g.f. A(x) is -A(-x).
%H Vaclav Kotesovec, <a href="/A089796/b089796.txt">Table of n, a(n) for n = 1..550</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1807.05794">Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences</a>, arXiv:1807.05794 [math.CO], 2018.
%F G.f. A(x)=y satisfies (x-x^2-x^3)+y*(-1+3*x-x^3)+y^2*(-1+x^2-x^3)+y^3*(1-x+x^2-x^3)=0.
%F a(n) ~ sqrt((-1 - 3*s + s^3 - 2*r*(-1 + s^2 + s^3) + 3*r^2*(1 + s + s^2 + s^3)) / (1 - 3*s + 3*r*s - r^2*(1 + 3*s) + r^3*(1 + 3*s))) / (2*sqrt(Pi) * n^(3/2) * r^(n-1/2)), where r = 0.3637032853572807454... and s = 0.8232781794881572707... are roots of the system of equations 3*r = 1 + r^3 + 2*(1 - r^2 + r^3)*s + 3*(-1 + r - r^2 + r^3)*s^2, s + s^2 + r*(-1 - 3*s + s^3) + r^3*(1 + s + s^2 + s^3) = s^3 + r^2*(-1 + s^2 + s^3). - _Vaclav Kotesovec_, Mar 10 2014
%t terms = 32; A[_] = 0;
%t Do[A[x_] = (x-x^2-x^3) + (3x-x^3) A[x] + (-1+x^2-x^3) A[x]^2 + (1-x+x^2-x^3) A[x]^3 + O[x]^(terms+1) // Normal, {terms+1}];
%t CoefficientList[A[x]/x, x] (* _Jean-François Alcover_, Nov 10 2018 *)
%o (PARI) a(n)=local(A); if(n<1,0,A=O(x); for(k=1,n,A=(x-x^2-x^3)+A*(3*x-x^3)+A^2*(-1+x^2-x^3)+A^3*(1-x+x^2-x^3)); polcoeff(A,n))
%Y a(n) = A006196(n) if n<28.
%K nonn
%O 1,4
%A _Michael Somos_, Nov 10 2003
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