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

0,5

Links

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

Formula

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

Example

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

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);

Crossrefs

Column k=9 of A243827.

Sequence in context: A055419 A027091 A183712 * A081495 A191645 A146240

Adjacent sequences: A244232 A244233 A244234 * A244236 A244237 A244238

Keyword

nonn

Author

Alois P. Heinz, Jun 23 2014

Status

approved