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%I #17 May 11 2013 07:57:43
%S 1,1,11,31,211,851,4901,22961,124531,623011,3313201,17086301,90453661,
%T 473616781,2509264811,13250049551,70368250451,373539254611,
%U 1989045489281,10597110956861,56566637447401,302196871378601,1616570627763311,8654955238504531,46384344189261661
%N G.f.: 1/(1-2x-19x^2)^(1/2).
%C Central coefficient of (1+x+5x^2)^n.
%C Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps can have five colors. - _N-E. Fahssi_, Mar 31 2008
%H Vincenzo Librandi, <a href="/A098264/b098264.txt">Table of n, a(n) for n = 0..300</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(x)*BesselI(0, 2*sqrt(5)x).
%F a(n) = sum{k=0..floor(n/2), binomial(n, k)*binomial(n-k, k)*5^k}.
%F a(n) = sum{k=0..floor(n/2), binomial(n, 2k)*binomial(2k, k)*5^k}.
%F n*a(n) +(1-2*n)*a(n-1) +19*(1-n)*a(n-2)=0. - _R. J. Mathar_, Nov 14 2011
%F a(n) ~ sqrt(50+5*sqrt(5))*(1+2*sqrt(5))^n/(10*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 14 2012
%t Table[SeriesCoefficient[1/Sqrt[1-2*x-19*x^2],{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%o (PARI) x='x+O('x^66); Vec(1/(1-2*x-19*x^2)^(1/2)) \\ _Joerg Arndt_, May 11 2013
%Y Cf. A084601, A084603, A084605.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Aug 31 2004
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