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A091565 Expansion of (1 - x - sqrt(1 - 2*x + x^2 - 8*x^3)) / (4*x^2) in powers of x. 0

%I #16 May 13 2022 05:51:07

%S 0,1,1,1,3,7,13,29,71,163,377,913,2219,5375,13189,32677,81167,202523,

%T 508273,1280537,3236275,8207543,20880893,53263405,136205527,349137811,

%U 896881641,2308523809,5953138875,15378562415,39791453685,103115768885

%N Expansion of (1 - x - sqrt(1 - 2*x + x^2 - 8*x^3)) / (4*x^2) in powers of x.

%C Series reversion of g.f. A(x) is -A(-x).

%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.: (1 - x - sqrt(1 - 2*x + x^2 - 8*x^3)) / (4*x^2) = 2*x / (1 - x + sqrt(1 - 2*x + x^2 - 8*x^3)) .

%F G.f. A(x) = y satisfies 2*(x*y)^2 + (x - 1)*y + x = 0.

%F a(n) = a(n-1) + 2*(a(1)*a(n-3) + a(2)*a(n-4) + ... + a(n-3)*a(1)) for n>1.

%F D-finite with recurrence: +(n+2)*a(n) +(-2*n-1)*a(n-1) +(n-1)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - _R. J. Mathar_, Jan 25 2020

%e G.f. = x + x^2 + x^3 + 3*x^4 + 7*x^5 + 13*x^6 + 29*x^7 + 71*x^8 + 163*x^9 + ...

%o (PARI) {a(n) = if( n<0, 0, polcoeff( 2 * x / (1 - x + sqrt(1 - 2*x + x^2 - 8*x^3 + x * O(x^n))), n))};

%Y Cf. A025249.

%K nonn

%O 0,5

%A _Michael Somos_, Jan 21 2004

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)