A114465 Number of Dyck paths of semilength n having no ascents of length 2 that start at an odd level.

1, 1, 2, 5, 13, 36, 105, 317, 982, 3105, 9981, 32520, 107157, 356481, 1195662, 4038909, 13728369, 46919812, 161143157, 555857157, 1924956954, 6689953057, 23325404153, 81567552320, 286009944649, 1005371062561, 3542175587306

Offset

0,3

Comments

Column 0 of A114463.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Jean-Luc Baril and Paul Barry, Two kinds of partial Motzkin paths with air pockets, arXiv:2212.12404 [math.CO], 2022.

Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, Consecutive pattern-avoidance in Catalan words according to the last symbol, Univ. Bourgogne (France 2023).

Jean-Luc Baril, Rigoberto Flórez, and José L. Ramírez, Counting symmetric and asymmetric peaks in motzkin paths with air pockets, Univ. Bourgogne (France, 2023).

Formula

G.f.: [1 - z^2 - sqrt((1+z^2)*(1-4z+z^2))]/[2*z*(1-z+z^2)].

(n+1)*a(n) = (5*n-1)*a(n-1) - (7*n-5)*a(n-2) + 10*(n-2)*a(n-3) - (7*n-23)*a(n-4) + (5*n-19)*a(n-5) - (n-5)*a(n-6). - Vaclav Kotesovec, Mar 20 2014

a(n) ~ sqrt(24+14*sqrt(3)) * (2+sqrt(3))^n / (6 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

Example

a(4)=13 because among the 14 Dyck paths of semilength 4 only UUD(UU)DDD has an ascent of length 2 that starts at an odd level (shown between parentheses).

Maple

g:=-1/2/z/(1+z^2-z)*(z^2-1+sqrt((z^2+1)*(z^2-4*z+1))): gser:=series(g, z=0, 33): 1, seq(coeff(gser, z^n), n=1..30);

Mathematica

CoefficientList[Series[(1-x^2-Sqrt[(1+x^2)*(1-4*x+x^2)])/(2*x*(1-x+x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

Prog

(PARI) Vec((1 - x^2 - sqrt((1+x^2)*(1-4*x+x^2)))/(2*x*(1-x+x^2)) + O(x^50)) \\ G. C. Greubel, Jan 28 2017

Crossrefs

Cf. A114463, A114462, A114464.

Sequence in context: A087626 A125094 A271941 * A135310 A135337 A133365

Adjacent sequences: A114462 A114463 A114464 * A114466 A114467 A114468

Keyword

nonn

Author

Emeric Deutsch, Nov 29 2005

Status

approved