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%I #40 Sep 10 2021 18:33:00
%S 1,3,3,-27,-189,-567,189,11259,59859,129033,-395847,-4730481,
%T -19580211,-21264201,258785523,1917734373,6051991059,-2659507911,
%U -135544952151,-738957668337,-1618780564359,5297724346923,63513121347063,266379249285873,286776522444861,-3683959713627417
%N Expansion of 1/sqrt(1 - 6x + 21x^2).
%C Binomial transform of A012000. Second binomial transform of A098333.
%C Central coefficients of (1 + 3x - 3x^2)^n.
%H Seiichi Manyama, <a href="/A098340/b098340.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(3*x)*BesselI(0, 2*sqrt(-3)*x).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*3^n*(-3)^(-k).
%F a(n) = 3^n*Sum{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)*(-3)^(-k).
%F D-finite with recurrence: n*a(n) + 3*(1-2*n)*a(n-1) + 21*(n-1)*a(n-2) = 0. - _R. J. Mathar_, Sep 26 2012
%F Lim sup n->infinity |a(n)|^(1/n) = sqrt(21). - _Vaclav Kotesovec_, Sep 29 2013
%F a(n) = 3^n*hypergeom([1/2 - n/2, -n/2], [1], -4/3). - _Peter Luschny_, Mar 19 2018
%p a := n -> 3^n*hypergeom([1/2 - n/2, -n/2], [1], -4/3):
%p seq(simplify(a(n)), n=0..21); # _Peter Luschny_, Mar 19 2018
%t CoefficientList[Series[1/Sqrt[1-6x+21x^2],{x,0,30}],x] (* _Harvey P. Dale_, Oct 07 2012 *)
%o (PARI) my(x = 'x + O('x^30)); Vec(1/sqrt(1-6*x+21*x^2)) \\ _Jinyuan Wang_, Sep 08 2019
%Y Cf. A012000, A098333.
%K easy,sign
%O 0,2
%A _Paul Barry_, Sep 03 2004