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%I #46 Sep 10 2021 18:34:53
%S 1,1,-3,-11,1,81,141,-363,-1791,-479,13597,29877,-54911,-353807,
%T -223443,2539989,6806529,-8302527,-73999299,-73313931,489731841,
%U 1584548241,-1110170163,-15812965611,-21391839999,94696016481
%N Expansion of 1/sqrt(1 - 2*x + 9*x^2).
%C Central coefficients of (1 + x - 2*x^2)^n.
%C Binomial transform of 1/sqrt(1+8*x^2), or (1,0,-4,0,24,0,...).
%C Binomial transform is A098336.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, ex. 7.56, p. 575.
%H G. C. Greubel, <a href="/A098332/b098332.txt">Table of n, a(n) for n = 0..1000</a>
%H Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
%H Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%H Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%F E.g.f.: exp(x)*BesselI(0, 2*sqrt(-2)*x);
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * binomial(2*k,k) * (-2)^k.
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n-k,k) * (-2)^k.
%F a(n) = (-1)^n * Sum_{k=0..n} binomial(n,k)^2 * (-2)^k.
%F G.f.: A(x) = 1/(2*T(0)+3*x-1) where T(k) = 1 - 2*x/(1 + x/T(k+1)); (continued fraction, 2-step ). - _Sergei N. Gladkovskii_, Aug 23 2012
%F D-finite with recurrence: a(n+2) = ((2*n+3)*a(n+1))/(n+2) - (9*(n+1)*a(n))/(n+2) with a(0)=1, a(1)=1. (See Graham, Knuth, and Patashnik). - _Alexander R. Povolotsky_, Aug 23 2012
%F a(n) = hypergeom([1/2-n/2, -n/2], [1], -8). - _Peter Luschny_, Sep 18 2014
%F a(n) = (3/2)*(9/2)^n*Sum_{k >= 0} (-1/2)^k*binomial(n+k,k)^2. - _Peter Bala_, Mar 02 2017
%p a := n -> hypergeom([1/2-n/2, -n/2], [1], -8);
%p seq(round(evalf(a(n),99)),n=0..30); # _Peter Luschny_, Sep 18 2014
%t Table[(-3)^n*LegendreP[n,-1/3],{n,0,20}] (* _Vaclav Kotesovec_, Jul 23 2013 *)
%t CoefficientList[Series[1/Sqrt[1 - 2*x + 9*x^2], {x,0,50}], x] (* _G. C. Greubel_, Feb 18 2017 *)
%o (PARI) x='x+O('x^25); Vec(1/sqrt(1 - 2*x + 9*x^2)) \\ _G. C. Greubel_, Feb 18 2017
%Y Cf. A126869, A012000, A116091, A098341.
%K easy,sign
%O 0,3
%A _Paul Barry_, Sep 03 2004
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