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
KEYWORD
easy,nonn,nice
AUTHOR
Cyril Banderier, Oct 04 2000
STATUS
approved