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%I #17 Jan 29 2024 03:26:01
%S 1,10,68,394,2092,10516,50920,239962,1107836,5033020,22572376,
%T 100168260,440604088,1923626344,8344694224,35998921978,154546983580,
%U 660652406572,2813422792696,11940478362796,50522190460072
%N Expansion of (1-z-sqrt(1-4z))/(1-4z)^2.
%H Vincenzo Librandi, <a href="/A104598/b104598.txt">Table of n, a(n) for n = 1..200</a>
%H D. Merlini, <a href="https://doi.org/10.46298/dmtcs.3323">Generating functions for the area below some lattice paths</a>, Discrete Mathematics and Theoretical Computer Science AC, 2003, 217-228.
%F Recurrence: n*(3*n-4)*a(n) = 2*(12*n^2-13*n-2)*a(n-1) - 8*(2*n-1)*(3*n-1) * a(n-2). - _Vaclav Kotesovec_, Oct 17 2012
%F a(n) ~ 3*2^(2*n-2)*n*(1-8/(3*sqrt(Pi)*sqrt(n))). - _Vaclav Kotesovec_, Oct 17 2012
%F a(n) = 2^(2*n-2)*(3*n+4)-(n+1)*C(2*n+1,n). - _Vaclav Kotesovec_, Oct 28 2012
%t Rest[CoefficientList[Series[(1-x-Sqrt[1-4*x])/(1-4*x)^2, {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%t Table[2^(2*n-2)*(3*n+4)-(n+1)*Binomial[2*n+1,n],{n,1,20}] (* _Vaclav Kotesovec_, Oct 28 2012 *)
%K nonn
%O 1,2
%A _Ralf Stephan_, Mar 17 2005