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Area under Motzkin paths.
1

%I #15 Aug 23 2018 04:28:48

%S 1,10,54,242,979,3728,13627,48382,168069,574040,1934346,6446824,

%T 21290563,69771854,227150074,735316478,2368536349,7596077384,

%U 24267094264,77258501372,245204480443,776060212130,2449968185161

%N Area under Motzkin paths.

%C a(n) is 2*the sum of areas under all Motzkin excursions of length n. (nonnegative walks beginning in 0, with jumps -1,0,+1)

%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.: -(3*x^3-x^2-7*x+1+sqrt((x+1)*(1-3*x))*(3*x^2+6*x-1)) / (2*(x+1) * (3*x-1)^2*x).

%F -(n+1)*(14*n-107)*a(n) +(170*n^2-1113*n+214)*a(n-1) +2*(-214*n^2+1239*n-638)*a(n-2) +6*(-66*n^2+611*n-991)*a(n-3) +9*(138*n^2-899*n+1257)*a(n-4) +27*(38*n-147)*(n-4)*a(n-5)=0. - _R. J. Mathar_, Aug 23 2018

%t f[x_] := -2*(3*x^3-x^2-7*x+1+Sqrt[(x+1)*(1-3*x)]*(3*x^2+6*x-1)) / (4*(x+1)*(3*x-1)^2*x); CoefficientList[ Series[ f[x], {x, 1, 23}], x] (* _Jean-François Alcover_, Dec 21 2011, from area sum g.f. *)

%K easy,nonn,nice

%O 1,2

%A _Cyril Banderier_, Oct 04 2000