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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 (list; graph; refs; listen; history; text; internal format)
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)*(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

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Last modified December 5 11:47 EST 2021. Contains 349557 sequences. (Running on oeis4.)