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%I #20 Jan 03 2020 03:07:02
%S 1,2,4,9,22,56,146,388,1048,2869,7942,22192,62510,177308,506008,
%T 1451866,4185788,12119696,35227748,102753800,300672368,882373261,
%U 2596389190,7658677856,22642421206,67081765932,199128719896,592179010350
%N Expansion of (1-2x-sqrt(1-4x+4x^2-4x^3))/(2x^2).
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%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.
%H Paul Barry, <a href="https://arxiv.org/abs/1910.00875">Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials</a>, arXiv:1910.00875 [math.CO], 2019.
%F G.f.: (1-2x-sqrt(1-4x+4x^2-4x^3))/(2x^2).
%F a(n) = 2*a(n-1)+a(1)*a(n-3)+a(2)*a(n-4)+...+a(n-3)*a(1) for n>1.
%F Series reversion of g.f. A(x) is -A(-x).
%F G.f. A(x) satisfies 0=f(x, A(x)) where f(x, y)=(xy)^2+2(xy)-(y-x).
%F Conjecture: (n+2)*a(n) -2*(2*n+1)*a(n-1) +4*(n-1)*a(n-2) +2*(5-2*n)*a(n-3)=0. - _R. J. Mathar_, Aug 14 2012
%t CoefficientList[Series[(1-2x-Sqrt[1-4x+4x^2-4x^3])/(2x^2),{x,0,30}],x] (* _Harvey P. Dale_, Jan 31 2015 *)
%o (PARI) a(n)=polcoeff((1-2*x-sqrt(1-4*x+4*x^2-4*x^3+x^3*O(x^n)))/2,n+2)
%Y Cf. A025247, A025265.
%K nonn
%O 1,2
%A _Michael Somos_, Jan 20 2004
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