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A057586 Area under Motzkin paths. 1
1, 10, 54, 242, 979, 3728, 13627, 48382, 168069, 574040, 1934346, 6446824, 21290563, 69771854, 227150074, 735316478, 2368536349, 7596077384, 24267094264, 77258501372, 245204480443, 776060212130, 2449968185161 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is 2*the sum of areas under all Motzkin excursions of length n. (nonnegative walks beginning in 0, with jumps -1,0,+1)

LINKS

Table of n, a(n) for n=1..23.

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

FORMULA

G.f.: -(3*x^3-x^2-7*x+1+sqrt((x+1)*(1-3*x))*(3*x^2+6*x-1)) / (2*(x+1) * (3*x-1)^2*x).

-(n+1)*(14*n-107)*a(n) +(170*n^2-1113*n+214)*a(n-1) +2*(-214*n^2+1239*n-638)*a(n-2) +6*(-66*n^2+611*n-991)*a(n-3) +9*(138*n^2-899*n+1257)*a(n-4) +27*(38*n-147)*(n-4)*a(n-5)=0. - R. J. Mathar, Aug 23 2018

MATHEMATICA

f[x_] := -2*(3*x^3-x^2-7*x+1+Sqrt[(x+1)*(1-3*x)]*(3*x^2+6*x-1)) / (4*(x+1)*(3*x-1)^2*x); CoefficientList[ Series[ f[x], {x, 1, 23}], x] (* Jean-Fran├žois Alcover, Dec 21 2011, from area sum g.f. *)

CROSSREFS

Sequence in context: A036600 A058645 A170940 * A213120 A198770 A222539

Adjacent sequences:  A057583 A057584 A057585 * A057587 A057588 A057589

KEYWORD

easy,nonn,nice

AUTHOR

Cyril Banderier, Oct 04 2000

STATUS

approved

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Last modified November 13 17:34 EST 2019. Contains 329106 sequences. (Running on oeis4.)