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%I #10 Jul 16 2014 10:32:44
%S 0,0,0,1,5,17,54,177,594,1997,6698,22487,75701,255455,863576,2923806,
%T 9913448,33658109,114417190,389385699,1326522885,4523352061,
%U 15437800028,52730424194,180244620903,616546133055,2110330086114,7227665869122,24768041790134
%N Number of Dyck paths of semilength n having exactly one occurrence of the consecutive pattern UDDU.
%H Alois P. Heinz, <a href="/A244235/b244235.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) ~ c * ((1+sqrt(13+16*sqrt(2)))/2)^n / sqrt(n), where c = 0.09016594515129336503624934471608236212385331150935643095582327... . - _Vaclav Kotesovec_, Jul 16 2014
%e a(3) = 1: UUDDUD.
%e a(4) = 5: UDUUDDUD, UUDDUDUD, UUDDUUDD, UUDUDDUD, UUUDDUDD.
%p a:= proc(n) option remember; `if`(n<4, binomial(n, 3),
%p (2*(n-1)*(112*n^5-1220*n^4+5251*n^3-11122*n^2+11566*n-4764)*a(n-1)
%p +(n-2)*(560*n^5-5820*n^4+23159*n^3-44070*n^2+40253*n-14010)*a(n-2)
%p -6*(n-2)*(n-3)*(112*n^4-884*n^3+2437*n^2-2436*n+486)*a(n-3)
%p +23*(n-2)*(n-3)*(n-4)*(112*n^3-492*n^2+623*n-267)*a(n-4)) /
%p (n*(n-1)*(n-3)*(112*n^3-828*n^2+1943*n-1494)))
%p end:
%p seq(a(n), n=0..30);
%Y Column k=9 of A243827.
%K nonn
%O 0,5
%A _Alois P. Heinz_, Jun 23 2014
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