Search: a329679 -id:a329679
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A329678
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Number of excursions of length n with Motzkin-steps consisting only of consecutive steps UD and DH.
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+10
3
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1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0
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COMMENTS
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The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.
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LINKS
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FORMULA
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G.f.: 1 + t + t^2 + t^3.
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EXAMPLE
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We only have the following four excursions of this type: the empty walk, H, UD and UDH.
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CROSSREFS
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Cf. A329670, A329677, A329679 (other Motzkin excursions avoiding certain consecutive steps such that the sequence counting them has growth rate zero).
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KEYWORD
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nonn,walk,easy
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AUTHOR
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STATUS
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approved
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A329677
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Number of excursions of length n with Motzkin-steps consisting only of consecutive steps UH, HD, and DH.
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+10
2
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1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
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OFFSET
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0
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COMMENTS
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The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.
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LINKS
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FORMULA
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G.f.: 1 + t + t^3 + t^4.
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EXAMPLE
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We only have the following four excursions of this type: the empty walk, H, UHD and UHDH.
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MATHEMATICA
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PadRight[#, 105] &@ CoefficientList[Series[1 + x + x^3 + x^4, {x, 0, 105}], x] (* Michael De Vlieger, Dec 16 2019 *)
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CROSSREFS
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Cf. A329670, A329678, A329679 (other Motzkin excursions avoiding certain consecutive steps such that the sequence counting them has growth rate zero).
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KEYWORD
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nonn,walk,easy
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AUTHOR
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STATUS
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approved
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