A191317 Number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) having no UDU's, where U=(1,1) and D=(1,-1).

1, 1, 2, 3, 5, 8, 14, 23, 40, 67, 117, 198, 346, 590, 1032, 1769, 3096, 5328, 9329, 16103, 28205, 48801, 85500, 148216, 259733, 450952, 790387, 1374044, 2408653, 4191814, 7349019, 12801243, 22445281, 39127766, 68611494, 119687036, 209890344, 366348367, 642493426, 1121992447, 1967839835

Offset: 0

Author

Emeric Deutsch, Jun 01 2011

Keywords

• nonn

Comments

a(n) = A191316(n,0).

Addendum Jun 18 2011: (Start)

Also the number of length n left factors of Dyck paths having no DUD's.

Also number of dispersed Dyck paths with no DUD's. Example: a(4)=5 because we have UDHH, UUDD, HUDH, HHUD, and HHHH (here H = (1,0)). (End)

Examples

			a(4)=5 because we have HHHH, HHUD, HUDH, UDHH, and UUDD, where U=(1,1), D=(1,-1), and H=(1,0). (UDUD does not qualify.)
a(4)=5 because we have UDUU, UUDD, UUDU, UUUD, and UUUU (UDUD does not qualify).
		

Crossrefs

Cf. A191316.

Programs

Maple

g := ((sqrt(1-2*z^2-3*z^4)-1+2*z-z^2+2*z^3)*1/2)/(z*(1-2*z+z^2-z^3)): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);
# alternative, Jun 18 2011:
g := (2*(1+z^2))/((1-2*z)*(1+z^2)+sqrt((1+z^2)*(1-3*z^2))): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);

Formula

G.f.: ( sqrt(1-2*z^2-3*z^4) -1+2*z-z^2+2*z^3 )/ (2*z*(1-2*z+z^2-z^3)) = 2*(1+z^2) / ( (1-2*z)*(1+z^2)+sqrt((1+z^2)*(1-3*z^2)) ) .

D-finite with recurrence (n+1)*a(n) +2*(-n-1)*a(n-1) +(-n+5)*a(n-2) +3*(n-3)*a(n-3) +(-5*n+19)*a(n-4) +2*(4*n-17)*a(n-5) +3*(-n+5)*a(n-6) +3*(n-5)*a(n-7)=0. - R. J. Mathar, Jul 22 2022