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 A057585 Area under Motzkin excursions. 2

%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)*(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. *)

%K easy,nonn,nice

%O 1,3

%A _Cyril Banderier_, Oct 04 2000

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Last modified December 5 18:49 EST 2019. Contains 329768 sequences. (Running on oeis4.)