

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
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OFFSET

0,3


COMMENTS

Column 0 of A114463.


LINKS

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


FORMULA

G.f.: [1  z^2  sqrt((1+z^2)*(14z+z^2))]/[2*z*(1z+z^2)].
(n+1)*a(n) = (5*n1)*a(n1)  (7*n5)*a(n2) + 10*(n2)*a(n3)  (7*n23)*a(n4) + (5*n19)*a(n5)  (n5)*a(n6).  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^2z)*(z^21+sqrt((z^2+1)*(z^24*z+1))): gser:=series(g, z=0, 33): 1, seq(coeff(gser, z^n), n=1..30);


MATHEMATICA

CoefficientList[Series[(1x^2Sqrt[(1+x^2)*(14*x+x^2)])/(2*x*(1x+x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)


PROG

(PARI) Vec((1  x^2  sqrt((1+x^2)*(14*x+x^2)))/(2*x*(1x+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



