OFFSET
0,3
COMMENTS
a(n) is 2*the sum of the areas under all Dyck paths of length n.
The Dyck paths considered in this sequence always have height >= 0 but do not need to finish at height = 0. n is the total number of steps.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..3309 (terms n = 1..400 from T. D. Noe)
C. Banderier, Analytic combinatorics of random walks and planar maps, PhD Thesis, 2001.
FORMULA
G.f.: 2*x*(8*x^2+4*x-1-sqrt(1-4*x^2)*(4*x^2+4*x-1))/(4*(1-2*x)^2*(1+2*x)*x^2). - corrected by Vaclav Kotesovec, Sep 11 2013
Recurrence: (n+1)*(4*n^3 - 28*n^2 + 55*n - 27)*a(n) = 2*(8*n^3 - 48*n^2 + 52*n + 27)*a(n-1) + 4*(2*n - 1)*(4*n^3 - 24*n^2 + 29*n + 18)*a(n-2) - 16*(2*n - 3)*(2*n^2 - 8*n - 1)*a(n-3) - 16*(n-3)*(4*n^3 - 16*n^2 + 11*n + 4)*a(n-4). - Vaclav Kotesovec, Sep 11 2013
a(n) ~ 3*n*2^(n-1) * (1-4*sqrt(2)/(3*sqrt(Pi*n))). - Vaclav Kotesovec, Sep 11 2013
MAPLE
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, [1, 0],
add((p-> p+[0, p[1]*(2*y+i)])(b(x-1, y+i)), i=[-1, 1])))
end:
a:= n-> add(b(n, n-2*i), i=0..n/2)[2]:
seq(a(n), n=0..31); # Alois P. Heinz, Apr 01 2026
MATHEMATICA
f[x_] := 2*(8*x^2+4*x-1-Sqrt[1-4*x^2]*(4*x^2+4*x-1)) / (4*(1-2*x)^2*(1+2*x)*x^2); CoefficientList[ Series[ f[x], {x, 0, 27}], x] (* Jean-François Alcover, Dec 21 2011, after area sum g.f. multiplied by 2 *)
CROSSREFS
KEYWORD
easy,nonn,nice,changed
AUTHOR
Cyril Banderier, Oct 04 2000
STATUS
approved