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%I #22 Jan 29 2024 03:25:56
%S 1,4,12,34,84,212,488,1162,2580,5932,12888,28948,61992,136936,290256,
%T 633178,1331892,2877308,6016760,12897340,26843256,57175384,118545072,
%U 251163204,519103624,1094915512,2256939888,4742198632,9752832720
%N Total area below the lattice paths of length n defined by the rule [(0),(k)->(k-1)(k+1)] (Dyck paths).
%H G. C. Greubel, <a href="/A094893/b094893.txt">Table of n, a(n) for n = 1..1000</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+x-sqrt(1-4*x^2))/((1-2*x)*(1-4*x^2)).
%F a(n) ~ 3*n*2^(n-2) * (1-4*sqrt(2)/(3*sqrt(Pi*n))). - _Vaclav Kotesovec_, Mar 20 2014
%F D-finite with recurrence: n*(3*n-5)*a(n) +4*(-3*n+4)*a(n-1) +4*(-6*n^2+13*n-1)*a(n-2) +8*(6*n-5)*a(n-3) +16*(3*n-2)*(n-2)*a(n-4)=0. - _R. J. Mathar_, Aug 21 2018
%F D-finite with recurrence: n*a(n) -2*n*a(n-1) +4*(-2*n+3)*a(n-2) +8*(2*n-3)*a(n-3) +16*(n-3)*a(n-4) +32*(-n+3)*a(n-5)=0. - _R. J. Mathar_, Aug 21 2018
%t CoefficientList[ Series[(1 + x - Sqrt[1 - 4*x^2])/((1 - 2*x)*(1 - 4*x^2)), {x, 0, 30}], x] (* _Robert G. Wilson v_, Jun 15 2004 *)
%o (PARI) x='x+O('x^50); Vec((1+x-sqrt(1-4*x^2))/((1-2*x)*(1-4*x^2))) \\ _G. C. Greubel_, Feb 16 2017
%K nonn,easy
%O 1,2
%A Donatella Merlini (merlini(AT)dsi.unifi.it), Jun 16 2004
%E More terms from _Robert G. Wilson v_, Jun 16 2004
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