%I
%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
%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.
%F G.f.: (x^2 + 2*x  1 + (x+1)*sqrt((x+1)*(13*x)))/(2*(3*x1)*(x+1)*x^2).
%F Recurrence: (n2)*(n+2)*a(n) = (n+1)*(4*n7)*a(n1) + (2*n^2  3*n  8)*a(n2)  3*(n1)*(4*n5)*a(n3)  9*(n2)*(n1)*a(n4).  _Vaclav Kotesovec_, Sep 11 2013
%F a(n) ~ 3^(n+1)/4 * (12*sqrt(3)/sqrt(Pi*n)).  _Vaclav Kotesovec_, Sep 11 2013
%p G:= (x^2+2*x1+(x+1)*sqrt((x+1)*(13*x)))/(2*(3*x1)*(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*x1+(x+1)*Sqrt[(x+1)*(13*x)]) / (2*(3*x1)*(x+1)*x^2); Drop[ CoefficientList[ Series[ f[x], {x, 0, 26}], x], 1] (* _JeanFrançois Alcover_, Dec 21 2011, from g.f. *)
%K easy,nonn,nice
%O 1,3
%A _Cyril Banderier_, Oct 04 2000
