A166286 Number of Dyck paths with no UUU's and no DDD's, of semilength n having no peak plateaux (U=(1,1), D=(1,-1)).

1, 1, 2, 3, 5, 9, 17, 34, 70, 147, 313, 673, 1459, 3185, 6995, 15445, 34265, 76342, 170744, 383214, 862814, 1948299, 4411167, 10011973, 22775773, 51920833, 118593423, 271376295, 622047011, 1428128025, 3283679333, 7560750299

Offset

0,3

Comments

A peak plateau is a run of consecutive peaks that is preceded by an upstep U and followed by a down step D; a peak consists of an upstep followed by a downstep.

Links

Table of n, a(n) for n=0..31.

Formula

a(n) = A166285(n,0).

G.f.: G(z) satisfies G = 1 + zG + z^2*G + z^3*G[G - 1/(1-z)].

D-finite with recurrence (n+3)*a(n) +(-5*n-9)*a(n-1) +2*(4*n+3)*a(n-2) -4*n*a(n-3) +2*(-2*n+9)*a(n-5) +2*(4*n-21)*a(n-6) +4*(-n+6)*a(n-7)=0. - R. J. Mathar, Jul 22 2022

Example

a(3)=3 because we have UDUDUD, UDUUDD, and UUDDUD (UUDUDD is a peak plateau).

Maple

F := RootOf(G = 1+z*G+z^2*G+z^3*G*(G-1/(1-z)), G): Fser := series(F, z = 0, 35): seq(coeff(Fser, z, n), n = 0 .. 32);

Crossrefs

Cf. A166285.

Sequence in context: A213705 A295637 A061902 * A179807 A110113 A341960

Adjacent sequences: A166283 A166284 A166285 * A166287 A166288 A166289

Keyword

nonn

Author

Emeric Deutsch, Oct 12 2009

Status

approved