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%I #30 Oct 27 2023 02:46:09
%S 0,1,4,16,56,190,624,2014,6412,20219,63284,196938,610052,1882717,
%T 5792528,17776102,54433100,166374109,507710420,1547195902,4709218604,
%U 14318240578,43493134160,132003957436,400337992056,1213314272395,3674980475284,11124919273160
%N Area under Motzkin excursions.
%C a(n) is the sum of areas under all Motzkin excursions of length n (nonnegative walks beginning and ending in 0, with jumps -1,0,+1).
%H T. D. Noe, <a href="/A057585/b057585.txt">Table of n, a(n) for n = 1..400</a>
%H C. Banderier, <a href="http://algo.inria.fr/banderier/">Analytic combinatorics of random walks and planar maps</a>, PhD Thesis, 2001.
%H AJ Bu, <a href="https://arxiv.org/abs/2310.17026">Explicit Generating Functions for the Sum of the Areas Under Dyck and Motzkin Paths (and for Their Powers)</a>, arXiv:2310.17026 [math.CO], 2023.
%H AJ Bu and Doron Zeilberger, <a href="https://arxiv.org/abs/2305.09030">Using Symbolic Computation to Explore Generalized Dyck Paths and Their Areas</a>, arXiv:2305.09030 [math.CO], 2023.
%F G.f.: (x^2 + 2*x - 1 + (-x+1)*sqrt((x+1)*(1-3*x)))/(2*(3*x-1)*(x+1)*x^2).
%F Recurrence: (n-2)*(n+2)*a(n) = (n+1)*(4*n-7)*a(n-1) + (2*n^2 - 3*n - 8)*a(n-2) - 3*(n-1)*(4*n-5)*a(n-3) - 9*(n-2)*(n-1)*a(n-4). - _Vaclav Kotesovec_, Sep 11 2013
%F a(n) ~ 3^(n+1)/4 * (1-2*sqrt(3)/sqrt(Pi*n)). - _Vaclav Kotesovec_, Sep 11 2013
%p G:= (x^2+2*x-1+(-x+1)*sqrt((x+1)*(1-3*x)))/(2*(3*x-1)*(x+1)*x^2): Gser:=series(G,x=0,30): seq(coeff(Gser,x,n),n=1..26); # _Emeric Deutsch_, Apr 08 2007
%t f[x_] := (x^2+2*x-1+(-x+1)*Sqrt[(x+1)*(1-3*x)]) / (2*(3*x-1)*(x+1)*x^2); Drop[ CoefficientList[ Series[ f[x], {x, 0, 26}], x], 1] (* _Jean-François Alcover_, Dec 21 2011, from g.f. *)
%Y Cf. A001006, A333498.
%K easy,nonn,nice
%O 1,3
%A _Cyril Banderier_, Oct 04 2000
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