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%I #28 Jan 29 2024 03:25:44
%S 1,7,34,144,563,2095,7532,26410,90853,307893,1030886,3417450,11235151,
%T 36676453,119003432,384098710,1234016321,3948461521,12588083810,
%U 40001960362,126745795259,400532044957,1262690290868,3971944688584,12469123686533,39071957204695,122222999571622
%N Total area below the lattice paths of a given length defined by the rule [(0),(k)->(k-1)(k)(k+1)] (Motzkin paths).
%H Vincenzo Librandi, <a href="/A094891/b094891.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 G.f.: (1-sqrt(1-2*t-3*t^2))/((1-3*t)^2*(1+t)).
%F D-finite with recurrence: n*(2*n-3)*a(n) = 2*(4*n^2 - 3*n-3)*a(n-1) + 2*(2*n^2 - 12*n +3) *a(n-2) - 6*(n-1)*(4*n-1)*a(n-3) - 9*(n-2)*(2*n-1)*a(n-4). - _Vaclav Kotesovec_, Oct 24 2012
%F a(n) ~ 3^(n+1)*n/4*(1-4/(sqrt(3*Pi*n))). - _Vaclav Kotesovec_, Oct 24 2012
%p G:=(1-sqrt(1-2*t-3*t^2))/((1-3*t)^2*(1+t)): Gser:=series(G,t=0,30): seq(coeff(Gser,t^n),n=1..28); # _Emeric Deutsch_, Dec 16 2004
%t Rest[CoefficientList[Series[(1-Sqrt[1-2x-3x^2])/((1-3x)^2 (1+x)), {x,0,30}], x]] (* _Harvey P. Dale_, Oct 20 2011 *)
%o (PARI) x='x+O('x^66); Vec((1-sqrt(1-2*x-3*x^2))/((1-3*x)^2*(1+x))) \\ _Joerg Arndt_, May 11 2013
%K nonn,easy
%O 1,2
%A Donatella Merlini (merlini(AT)dsi.unifi.it), Jun 16 2004
%E More terms from _Emeric Deutsch_, Dec 16 2004
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