A114464 Number of Dyck paths of semilength n having no ascents of length 2 that start at an even level.
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
Offset: 0
Keywords
Examples
a(4)=6 because we have UDUDUDUD, UDUUUDDD, UUUDDDUD, UUUDUDDD, UUUDDUDD and UUUUDDDD, where U=(1,1), D=(1,-1).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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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);
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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 *)
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
Comments