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%I #47 Aug 29 2023 11:21:59
%S 1,1,3,5,13,27,67,153,375,893,2189,5319,13089,32155,79479,196573,
%T 487833,1212135,3018355,7525585,18792303,46980373,117589689,294613155,
%U 738844719,1854484305,4658460165,11710592711,29458662005,74151824271
%N a(n) = T(n,n), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=0.
%C Number of grand Motzkin n-paths avoiding UF. - _David Scambler_, Jun 20 2013
%H Vincenzo Librandi, <a href="/A026569/b026569.txt">Table of n, a(n) for n = 0..200</a>
%H Rui Duarte and António Guedes de Oliveira, <a href="https://www.cmup.pt/sites/default/files/2023-08/GF_LP_corrected_0.pdf">Generating functions of lattice paths</a>, Univ. do Porto (Portugal 2023).
%H J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%F a(n) = Sum_{k=0..floor(n/2)} binomial(2*k, k)*binomial(n-k, k). - _Paul Barry_, Sep 09 2004
%F G.f.: sqrt(1/((1-x)*(1-x-4*x^2))). - _Ralf Stephan_, Jan 08 2004
%F D-finite with recurrence: a(n) = 1/n*((2*n-1)*a(n-1) + (3*n-3)*a(n-2) - (4*n-6)*a(n-3)). - _Vladeta Jovovic_, Mar 12 2005
%F a(n) = Sum_{k=0..n} C(k, n-k)*C(2*(n-k), n-k). - _Paul Barry_, Jul 30 2005
%F G.f.: 1/(1-x-2*x^2/(1-0*x-x^2/(1-x-x^2/(1-0*x-2*x^2/(1-x-x^2/.... (continued fraction). _Paul Barry_, Dec 07 2008
%F a(n) ~ sqrt((5+13/sqrt(17))/8) * ((1+sqrt(17))/2)^n/sqrt(Pi*n). - _Vaclav Kotesovec_, Aug 10 2013
%e For a(3) = 5 the five grand Motzkin paths are FDU, DFU, FUD, UDF and FFF. The paths containing UF, namely UFD and DUF, are avoided. - _David Scambler_, Jun 20 2013
%t CoefficientList[Series[Sqrt[1/((1-x)(1-x-4x^2))],{x,0,30}],x] (* _Harvey P. Dale_, Oct 06 2011 *)
%o (PARI) my(x='x+O('x^30)); Vec( 1/sqrt((1-x)*(1-x-4*x^2)) ) \\ _G. C. Greubel_, Aug 03 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt((1-x)*(1-x-4*x^2)) )); // _G. C. Greubel_, Aug 03 2019
%o (Sage) (1/sqrt((1-x)*(1-x-4*x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 03 2019
%o (GAP) List([0..30], n-> Sum([0..Int(n/2)], k-> Binomial(2*k,k)*Binomial( n-k, k) )); # _G. C. Greubel_, Aug 03 2019
%Y Cf. A026568.
%K nonn
%O 0,3
%A _Clark Kimberling_
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