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A057585 Area under Motzkin excursions. 2
0, 1, 4, 16, 56, 190, 624, 2014, 6412, 20219, 63284, 196938, 610052, 1882717, 5792528, 17776102, 54433100, 166374109, 507710420, 1547195902, 4709218604, 14318240578, 43493134160, 132003957436, 400337992056, 1213314272395 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..400

C. Banderier, Analytic combinatorics of random walks and planar maps, PhD Thesis, 2001.

FORMULA

G.f.: (x^2 + 2*x - 1 + (-x+1)*sqrt((x+1)*(1-3*x)))/(2*(3*x-1)*(x+1)*x^2).

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

a(n) ~ 3^(n+1)/4 * (1-2*sqrt(3)/sqrt(Pi*n)). - Vaclav Kotesovec, Sep 11 2013

MAPLE

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

MATHEMATICA

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-Fran├žois Alcover, Dec 21 2011, from g.f. *)

CROSSREFS

Sequence in context: A026126 A026155 A025182 * A255301 A097128 A006079

Adjacent sequences:  A057582 A057583 A057584 * A057586 A057587 A057588

KEYWORD

easy,nonn,nice

AUTHOR

Cyril Banderier, Oct 04 2000

STATUS

approved

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Last modified October 19 13:01 EDT 2019. Contains 328222 sequences. (Running on oeis4.)