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A114464 Number of Dyck paths of semilength n having no ascents of length 2 that start at an even level. 6
1, 1, 1, 2, 6, 18, 54, 166, 522, 1670, 5418, 17786, 58974, 197226, 664494, 2253390, 7685394, 26345230, 90721362, 313682098, 1088609142, 3790610306, 13239554790, 46371693174, 162835695258, 573160873750, 2021885799162, 7146955776554 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Column 0 of A114462.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

G.f.=[1-z+3z^2-z^3-(1-z)sqrt((1-4z+z^2)(1+z^2))]/(2z).

G.f. 1+x/(1-x)c(x^2/(1-x)^4), c(x) the g.f. of A000108; a(n+1)=sum{k=0..floor(n/2), C(n+2k,4k)C(k)}; - Paul Barry, May 31 2006

Conjecture: (n+1)*a(n) +(-5*n+3)*a(n-1) +2*(3*n-7)*a(n-2) +2*(-3*n+11)*a(n-3) +(5*n-27)*a(n-4) +(-n+7)*a(n-5)=0. - R. J. Mathar, Nov 26 2012

Recurrence: (n-3)*(n+1)*a(n) = (4*n^2 - 14*n + 9)*a(n-1) - (2*n^2 - 10*n + 15)*a(n-2) + (4*n^2 - 26*n + 39)*a(n-3) - (n-6)*(n-2)*a(n-4). - Vaclav Kotesovec, Feb 13 2014

a(n) ~ sqrt(2*sqrt(3)-3) * (2+sqrt(3))^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014

EXAMPLE

a(4)=6 because we have UDUDUDUD, UDUUUDDD, UUUDDDUD, UUUDUDDD, UUUDDUDD and UUUUDDDD, where U=(1,1), D=(1,-1).

MAPLE

G:=(1-z+3*z^2-z^3-(1-z)*sqrt((1-4*z+z^2)*(1+z^2)))/2/z: Gser:=series(G, z=0, 33): 1, seq(coeff(Gser, z^n), n=1..30);

MATHEMATICA

CoefficientList[Series[(1-x+3*x^2-x^3-(1-x)*Sqrt[(1-4*x+x^2)*(1+x^2)])/2/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

CROSSREFS

Cf. A114462, A114463, A114465.

Sequence in context: A008776 A134635 A192338 * A062415 A086680 A275872

Adjacent sequences:  A114461 A114462 A114463 * A114465 A114466 A114467

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 29 2005

STATUS

approved

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Last modified October 17 11:18 EDT 2021. Contains 348048 sequences. (Running on oeis4.)