%I #50 Dec 13 2018 08:50:15

%S 1,1,5,13,49,161,581,2045,7393,26689,97285,355565,1305745,4808545,

%T 17760965,65753693,243954113,906758785,3375949829,12587460557,

%U 46995614449,175669746209,657370655045,2462383495357,9232029156001

%N Coefficients of 1/(1-2x-7x^2)^(1/2); also, a(n) is the central coefficient of (1+x+2x^2)^n.

%C The Hankel transform (see A001906 for definition) of this sequence is A036442 : 1, 4, 32, 512, 16384, ... . - _Philippe Deléham_, Jul 03 2005

%C Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), U (or D) can have 2 colors. - _N-E. Fahssi_, Feb 05 2008

%H Seiichi Manyama, <a href="/A084601/b084601.txt">Table of n, a(n) for n = 0..1718</a> (terms 0..120 from Vincenzo Librandi)

%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). - _Vladeta Jovovic_, Mar 21 2004

%F a(n)=sum{k=0..floor(n/2), binomial(n-k, k)binomial(n, k)2^k}. - _Paul Barry_, Aug 26 2004

%F Sum[k=0..n, Trinomial(k, n) Binomial(n, k) ], with Trinomial=A027907. - _Ralf Stephan_, Jan 28 2005

%F a(n) is also the central coefficient of (2+x+x^2)^n; a(n)=sum_{k=0..n} C(n,k) T(k,n), where T(k,n) is the triangle of trinomial coefficients = coefficient of x^n of (1+x+x^2)^k : A027907 - _N-E. Fahssi_, Feb 05 2008

%F a(n+2)=( (2*n+3)*a(n+1) + 7*(n+1)*a(n) )/(n+2); a(0)=a(1)=1 - Sergei N. Gladkovskii, Aug 01 2012

%F G.f.: G(0), where G(k)= 1 + x*(2+7*x)*(4*k+1)/( 4*k+2 - x*(2+7*x)*(4*k+2)*(4*k+3)/(x*(2+7*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 06 2013

%F a(n) ~ sqrt(8+2*sqrt(2)) * (1+2*sqrt(2))^n / (4*sqrt(Pi*n)). - _Vaclav Kotesovec_, May 09 2014

%F a(n) = hypergeom([1/2 - n/2, -n/2], [1], 8). - _Peter Luschny_, Mar 18 2018

%p a := n -> hypergeom([1/2 - n/2, -n/2], [1], 8):

%p seq(simplify(a(n)), n=0..24); # _Peter Luschny_, Mar 18 2018

%t CoefficientList[Series[1/Sqrt[1-2x-7x^2],{x,0,30}],x] (* _Harvey P. Dale_, Sep 18 2011 *)

%o (PARI) for(n=0,30,t=polcoeff((1+x+2*x^2)^n,n,x); print1(t","))

%o (Maxima) a(n):=coeff(expand((1+x+2*x^2)^n),x,n);

%o makelist(a(n),n,0,12); /* Emanuele Munarini, Mar 02 2011 */

%Y Cf. A002426, A084600, A084602-A084615.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 01 2003