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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 UDUs, where U=(1,1) and D=(1,-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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

Addendum Jun 18 2011:

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)

LINKS

Table of n, a(n) for 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)) ) .

EXAMPLE

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).

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);

CROSSREFS

Cf. A191316

Sequence in context: A120400 A217283 A000621 * A218020 A039828 A246360

Adjacent sequences:  A191314 A191315 A191316 * A191318 A191319 A191320

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jun 01 2011

STATUS

approved

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Last modified December 7 14:52 EST 2016. Contains 278877 sequences.