%I #15 Sep 23 2014 04:26:57
%S 0,1,5,29,185,1257,8925,65445,491825,3768209,29324405,231153133,
%T 1841801065,14810069497,120029657805,979470140661,8040831465825,
%U 66361595715105,550284185213925,4582462506008253,38306388126997785
%N Series reversion of x/(1+5x+4x^2).
%C Hankel transform is -A127847(n)=-4^C(n,2)*(4^n-1)/3; a(n+1) counts (5,4)-Motzkin paths of length n, where there are 5 colors available for the H(1,0) steps and 4 for the U(1,1) steps. See A059231 for more information.
%H Vincenzo Librandi, <a href="/A127846/b127846.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: (1-5x-sqrt(1-10x+9x^2))/(8x); a(n)=sum{k=0..n-1, (1/n)*C(n,k)C(n,k+1)4^k}; a(n+1)=sum{k=0..floor(n/2), C(n, 2k)C(k)5^(n-2k)*4^k};
%F Recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - 9*(n-2)*a(n-2). - _Vaclav Kotesovec_, Oct 19 2012
%F a(n) ~ 3^(2*n+1)/(4*sqrt(2*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 19 2012
%F a(n) = A059231(n) for n>0. - _Philippe Deléham_, Apr 03 2013
%F a(n) = hypergeom([1-n, -n], [2], 4) for n>0. - _Peter Luschny_, Sep 23 2014
%t CoefficientList[Series[(1-5*x-Sqrt[1-10*x+9*x^2])/(8*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 19 2012 *)
%o (Sage)
%o A127846 = lambda n: hypergeometric([1-n, -n], [2], 4) if n>0 else 0
%o [Integer(A127846(n).n(100)) for n in (0..22)] # _Peter Luschny_, Sep 23 2014
%Y Cf. A059231
%K easy,nonn
%O 0,3
%A _Paul Barry_, Feb 02 2007