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%I #29 Jan 30 2020 21:29:15
%S 1,3,7,9,-21,-207,-911,-2769,-5213,2457,74997,400491,1409109,3323583,
%T 2219343,-27453951,-186624333,-750905127,-2088947819,-2955863589,
%U 8506703569,86421384387,401183114163,1280139325101,2522745571021
%N Expansion of 1/sqrt(1-6x+13x^2).
%C Binomial transform of A098335. Second binomial transform of A098331.
%C Central coefficients of (1+3x-x^2)^n.
%H Robert Israel, <a href="/A098338/b098338.txt">Table of n, a(n) for n = 0..1796</a>
%H Hacène Belbachir, Abdelghani Mehdaoui, 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 Robert Israel, <a href="/A098338/a098338.png">Plot of a(n) sqrt(n)/13^(n/2) for 1<=n<=10000</a>.
%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*I*x), I=sqrt(-1).
%F a(n) = Sum{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*3^n*(-9)^(-k).
%F a(n) = Sum{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)*3^n*(-9)^(-k).
%F D-finite with recurrence: n*a(n) +3*(1-2*n)*a(n-1) +13*(n-1)*a(n-2)=0. - _R. J. Mathar_, Sep 26 2012
%F Recurrence follows from the differential equation (13x-3) g(x) + (13x^2-6x+1) g'(x) = 0 satisfied by the generating function. - _Robert Israel_, Mar 02 2017
%F Lim sup n->infinity |a(n)|^(1/n) = sqrt(13). - _Vaclav Kotesovec_, Sep 29 2013
%p f:= gfun:-rectoproc({(13*n+13)*a(n)+(-9-6*n)*a(n+1)+(n+2)*a(n+2), a(0)=1, a(1)=3},a(n),remember):
%p map(f, [$0..50]); # _Robert Israel_, Mar 02 2017
%t CoefficientList[Series[1/Sqrt[1-6*x+13*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 29 2013 *)
%K easy,sign
%O 0,2
%A _Paul Barry_, Sep 03 2004
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