A244235 Number of Dyck paths of semilength n having exactly one occurrence of the consecutive pattern UDDU.

0, 0, 0, 1, 5, 17, 54, 177, 594, 1997, 6698, 22487, 75701, 255455, 863576, 2923806, 9913448, 33658109, 114417190, 389385699, 1326522885, 4523352061, 15437800028, 52730424194, 180244620903, 616546133055, 2110330086114, 7227665869122, 24768041790134

Offset: 0

Author

Alois P. Heinz, Jun 23 2014

Keywords

• nonn

Examples

			a(3) = 1: UUDDUD.
a(4) = 5: UDUUDDUD, UUDDUDUD, UUDDUUDD, UUDUDDUD, UUUDDUDD.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=9 of A243827.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, binomial(n, 3),
     (2*(n-1)*(112*n^5-1220*n^4+5251*n^3-11122*n^2+11566*n-4764)*a(n-1)
     +(n-2)*(560*n^5-5820*n^4+23159*n^3-44070*n^2+40253*n-14010)*a(n-2)
     -6*(n-2)*(n-3)*(112*n^4-884*n^3+2437*n^2-2436*n+486)*a(n-3)
     +23*(n-2)*(n-3)*(n-4)*(112*n^3-492*n^2+623*n-267)*a(n-4)) /
     (n*(n-1)*(n-3)*(112*n^3-828*n^2+1943*n-1494)))
    end:
seq(a(n), n=0..30);

Formula

a(n) ~ c * ((1+sqrt(13+16*sqrt(2)))/2)^n / sqrt(n), where c = 0.09016594515129336503624934471608236212385331150935643095582327... . - Vaclav Kotesovec, Jul 16 2014