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A057571 Area under Dyck paths. 3
1, 6, 19, 58, 146, 380, 883, 2138, 4774, 11092, 24190, 54724, 117508, 260920, 554179, 1213690, 2557022, 5541092, 11601610, 24930860, 51942076, 110861896, 230053614, 488253348, 1009853116, 2133122760, 4399720348, 9256078408 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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.: 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

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-Fran├žois Alcover, Dec 21 2011, after area sum g.f. multiplied by 2 *)

CROSSREFS

Sequence in context: A274599 A286184 A027044 * A238055 A272227 A272587

Adjacent sequences:  A057568 A057569 A057570 * A057572 A057573 A057574

KEYWORD

easy,nonn,nice

AUTHOR

Cyril Banderier, Oct 04 2000

STATUS

approved

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Last modified October 17 16:42 EDT 2018. Contains 316286 sequences. (Running on oeis4.)