A057585 Area under Motzkin excursions.

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, 3674980475284, 11124919273160

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.

AJ Bu, Explicit Generating Functions for the Sum of the Areas Under Dyck and Motzkin Paths (and for Their Powers), arXiv:2310.17026 [math.CO], 2023.

AJ Bu and Doron Zeilberger, Using Symbolic Computation to Explore Generalized Dyck Paths and Their Areas, arXiv:2305.09030 [math.CO], 2023.

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-François Alcover, Dec 21 2011, from g.f. *)

Crossrefs

Cf. A001006, A333498.

Sequence in context: A026126 A026155 A025182 * A333107 A255301 A097128

Adjacent sequences: A057582 A057583 A057584 * A057586 A057587 A057588

Keyword

easy,nonn,nice

Author

Cyril Banderier, Oct 04 2000

Status

approved