%I #19 Jan 12 2020 11:27:51

%S 1,1,2,5,14,41,110,245,450,739,1126,1625,2250,3015,3934,5021,6290,

%T 7755,9430,11329,13466,15855,18510,21445,24674,28211,32070,36265,

%U 40810,45719,51006,56685,62770,69275,76214,83601,91450,99775,108590,117909,127746,138115,149030,160505,172554

%N Number of Dyck paths of semilength n avoiding the pattern U^4 D^4 U D.

%H Colin Barker, <a href="/A225691/b225691.txt">Table of n, a(n) for n = 0..1000</a>

%H Axel Bacher, Antonio Bernini, Luca Ferrari, Benjamin Gunby, Renzo Pinzani and Julian West, <a href="http://dx.doi.org/10.1016/j.disc.2013.12.011">The Dyck pattern poset</a> Discrete Math. 321 (2014), 12--23. MR3154009.

%H A. Bernini, L. Ferrari, R. Pinzani and J. West, <a href="http://arxiv.org/abs/1303.3785">The Dyck pattern poset</a>, arXiv preprint arXiv:1303.3785, 2013

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = (14*n^3-84*n^2+124*n-84)/6 for n >= 6.

%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>6. - _Colin Barker_, Jul 10 2015

%F G.f.: (10*x^9-20*x^8+12*x^6+8*x^5+3*x^4-x^3+4*x^2-3*x+1) / (x-1)^4. - _Colin Barker_, Jul 10 2015

%t LinearRecurrence[{4,-6,4,-1},{1,1,2,5,14,41,110,245,450,739},50] (* _Harvey P. Dale_, Apr 10 2019 *)

%o (PARI) Vec((10*x^9-20*x^8+12*x^6+8*x^5+3*x^4-x^3+4*x^2-3*x+1)/(x-1)^4 + O(x^100)) \\ _Colin Barker_, Jul 10 2015

%Y A row of A238095.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, May 27 2013