A162985 Number of Dyck paths with no UUU's and no DDD's of semilength n and having no UUDUDD's (U=(1,1), D=(1,-1)).

1, 1, 2, 3, 6, 12, 25, 53, 114, 249, 550, 1227, 2760, 6253, 14256, 32682, 75293, 174224, 404741, 943622, 2207135, 5177817, 12179904, 28722736, 67890481, 160812128, 381671061, 907529504, 2161622683, 5157014539, 12321750366, 29482362166

Offset

0,3

Comments

a(n) = A162984(n,0).

Links

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

Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Formula

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

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

Example

a(3)=3 because we have UDUDUD, UDUUDD, and UUDDUD.

Maple

G := ((1-z-z^2+z^3-sqrt(1-2*z-z^2-z^4-2*z^5+z^6))*1/2)/z^3: Gser := series(G, z = 0, 36): seq(coeff(Gser, z, n), n = 0 .. 31);

Crossrefs

Cf. A162984.

Sequence in context: A004111 A032235 A192805 * A052523 A262430 A204855

Adjacent sequences: A162982 A162983 A162984 * A162986 A162987 A162988

Keyword

nonn

Author

Emeric Deutsch, Oct 11 2009

Status

approved