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A263173 Number of Dyck paths of semilength n having exactly two (possibly overlapping) DUDU's (with U=(1,1), D=(1,-1)).

%I A263173 #6 Oct 12 2015 09:09:05
%S A263173 1,3,15,58,231,891,3403,12870,48318,180356,670014,2479302,9143885,
%T A263173 33627777,123366789,451612846,1650111453,6019100025,21922936343,
%U A263173 79740801036,289690000380,1051250045960,3811012240380,13802994382860,49950211130905,180617997397887
%N A263173 Number of Dyck paths of semilength n having exactly two (possibly overlapping) DUDU's (with U=(1,1), D=(1,-1)).
%H A263173 Alois P. Heinz, <a href="/A263173/b263173.txt">Table of n, a(n) for n = 4..1000</a>
%e A263173 a(4) = 1: UDUDUDUD.
%e A263173 a(5) = 3: UDUDUDUUDD, UUDDUDUDUD, UUDUDUDUDD.
%e A263173 a(6) = 15: UDUDUDUUDDUD, UDUDUDUUDUDD, UDUDUDUUUDDD, UDUDUUDDUDUD, UDUDUUDUDUDD, UDUUDDUDUDUD, UDUUDUDUDUDD, UUDDUDUDUUDD, UUDUDDUDUDUD, UUDUDUDDUDUD, UUDUDUDUDDUD, UUDUDUDUUDDD, UUUDDDUDUDUD, UUUDDUDUDUDD, UUUDUDUDUDDD.
%p A263173 a:= proc(n) option remember; `if`(n<5, `if`(n=4, 1, 0),
%p A263173        ((2*n-7)*a(n-1) +(5*n-15)*a(n-2) +(2*n-5)*a(n-3)
%p A263173         -(n-2)*a(n-4))/(n-4))
%p A263173     end:
%p A263173 seq(a(n), n=4..30);
%Y A263173 Column k=2 of A102405.
%K A263173 nonn
%O A263173 4,2
%A A263173 _Alois P. Heinz_, Oct 11 2015