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A191317
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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).
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1
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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
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OFFSET
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0,3
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COMMENTS
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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)
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LINKS
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Table of n, a(n) for n=0..40.
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FORMULA
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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)) ) .
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EXAMPLE
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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).
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MAPLE
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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);
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CROSSREFS
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Cf. A191316
Sequence in context: A120400 A217283 A000621 * A218020 A039828 A005627
Adjacent sequences: A191314 A191315 A191316 * A191318 A191319 A191320
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, Jun 01 2011
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STATUS
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approved
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