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%I #11 Dec 21 2024 00:49:56
%S -1,4,-11,30,-83,232,-654,1856,-5296,15180,-43675,126062,-364863,
%T 1058552,-3077533,8963862,-26151753,76409052,-223544241,654790218,
%U -1920055017,5635816776,-16557539124,48685404516,-143264248974,421879104836,-1243160223829,3665516301186
%N Expansion of 2*(x - 1)^3/(3*x^3 - 5*x^2 + x + 1 + sqrt(-(x - 1)^3*(x + 1)^2*(3*x + 1)))
%C Binomial transform of A057552(n)*(-1)^(n+1).
%F G.f. A(x) satisfies: (-3*x^3 - x^2)*A(x)^2 + (3*x^3 - 5*x^2 + x + 1)*A(x) + (-x^3 + x*y^2 - x*y + 1) = 0.
%F a(n) = Limit_{k->oo} (A378783(k, k-n) - A378783(k, k-n-1)).
%F a(n) = A025566(n+1)+A025566(n+2)*(-1)^(n+1), for n > 0.
%F a(n) = Sum_{k=1..n+1} binomial(n, k-1)*(-1)^k*Sum_{m=0..k-1} binomial(2*m+2, m).
%o (PARI)
%o a(n) = sum(k=1, n+1, binomial(n, k-1)*(-1)^k*sum(m=0, k-1, binomial(2*m+2, m)))
%Y Cf. A025566, A057552, A378783, A378816 ( Hankel sequence transform ).
%K sign
%O 0,2
%A _Thomas Scheuerle_, Dec 08 2024