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A114465
Number of Dyck paths of semilength n having no ascents of length 2 that start at an odd level.
6
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
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Nov 29 2005
STATUS
approved