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
Keywords
Examples
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).
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).
Programs
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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);
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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 *) -
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
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
Comments