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A244235 Number of Dyck paths of semilength n having exactly one occurrence of the consecutive pattern UDDU. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 30 21:27 EDT 2020. Contains 337440 sequences. (Running on oeis4.)