A191794 Number of length n left factors of Dyck paths having no UUDD's; here U=(1,1) and D=(1,-1).

1, 1, 2, 3, 5, 8, 14, 23, 41, 69, 124, 212, 383, 662, 1200, 2091, 3799, 6661, 12122, 21359, 38919, 68850, 125578, 222892, 406865, 724175, 1322772, 2360010, 4313155, 7711148, 14099524, 25252819, 46192483, 82863807, 151628090, 272385447, 498578411, 896774552

Offset: 0

Author

Emeric Deutsch, Jun 18 2011

Keywords

• nonn

Examples

			a(4)=5 because we have UDUU, UDUD, UUDU, UUUD, and UUUU, where U=(1,1) and D=(1,-1) (the path UUDD does not qualify).
		

Links

• Helmut Prodinger, Dispersed Dyck paths revisited, arXiv:2402.13026 [math.CO], 2024.

Crossrefs

Cf. A191793.

Programs

Maple

g := 2/(1-2*z+z^4+sqrt(1-4*z^2+2*z^4+z^8)): gser := series(g, z = 0, 42): seq(coeff(gser, z, n), n = 0 .. 37);

Mathematica

CoefficientList[Series[2/(1-2x+x^4+Sqrt[1-4x^2+2x^4+x^8]), {x,0,40}], x] (* Harvey P. Dale, Jun 19 2011 *)

Formula

a(n) = A191793(n,0).

G.f.: g(z) = 2/(1-2*z+z^4+sqrt(1-4*z^2+2*z^4+z^8)).

D-finite with recurrence (n+1)*a(n) +2*(-1)*a(n-1) +4*(-n+1)*a(n-2) +2*(n-3)*a(n-4) +6*a(n-5) +(n-7)*a(n-8)=0. - R. J. Mathar, Jul 22 2022