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Expansion of ((x-1)*sqrt(x^2-6*x+1)-x^2-4*x+1)/(8*x^3).
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%I #58 Sep 08 2022 08:46:07

%S 1,4,17,76,353,1688,8257,41128,207905,1063932,5501073,28695252,

%T 150827073,798054000,4247388417,22722717328,122125206977,659101365812,

%U 3570473750929,19407758970268,105820555054241,578622868877704,3172136074486337

%N Expansion of ((x-1)*sqrt(x^2-6*x+1)-x^2-4*x+1)/(8*x^3).

%H Vincenzo Librandi, <a href="/A239204/b239204.txt">Table of n, a(n) for n = 0..250</a>

%H Letong Hong and Rupert Li, <a href="https://arxiv.org/abs/2112.15081">Length-Four Pattern Avoidance in Inversion Sequences</a>, arXiv:2112.15081 [math.CO], 2021.

%F a(n) = (Sum_{i=0..n+3} binomial(n+1, n-i+3)*binomial(n+i, n) )/(2*(n+1)).

%F a(n) ~ sqrt(3*sqrt(2)-4) / (4*sqrt(Pi) * n^(3/2) * (sqrt(2)-1)^(2*n+5)). - _Vaclav Kotesovec_, Mar 20 2014

%F a(n) = hypergeom([-n,-n-2], [2], 2). - _Peter Luschny_, Sep 23 2014

%F a(n) = (A001003(n+2) - A001003(n+1))/2. - _Philippe Deléham_, Sep 23 2014

%F Conjectured to be D-finite with recurrence: (n+3)*a(n) + (-7*n-10)*a(n-1) + (7*n-3)*a(n-2) + (-n+2)*a(n-3) = 0. - _R. J. Mathar_, Nov 02 2014

%F From _Peter Bala_, Jan 28 2020: (Start)

%F O.g.f. A(x) satisfies A(x*(1 - x)/(1 + x)) = Sum_{n >= 0} (n + 1)^2*x^n.

%F Equivalently, A(x) = (1 + x*S(x))/(1 - x*S(x))^3 where S(x) is the o.g.f. for the large Schröder numbers A006318. (End)

%t CoefficientList[Series[((x-1)*Sqrt[x^2-6*x+1]-x^2-4*x+1)/(8*x^3), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)

%o (Maxima) a(n):=sum(binomial(n+1, n-i+3)*binomial(n+i,n),i,0,n+3)/(2*(n+1));

%o (Magma) m:=30; R<x>:=LaurentSeriesRing(RationalField(), m); Coefficients(R!(((x-1)*Sqrt(x^2-6*x+1)-x^2-4*x+1)/(8*x^3))); // _Bruno Berselli_, Mar 18 2014

%o (Sage)

%o A239204 = lambda n: hypergeometric([-n,-n-2],[2],2)

%o [Integer(A239204(n).n(100)) for n in range(22)] # _Peter Luschny_, Sep 23 2014

%Y Cf. A085403, A006318.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Mar 17 2014