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A225015
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Number of sawtooth patterns of length 1 in all Dyck paths of semilength n.
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2
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0, 1, 1, 5, 18, 66, 245, 918, 3465, 13156, 50193, 192270, 739024, 2848860, 11009778, 42642460, 165480975, 643281480, 2504501625, 9764299710, 38115568260, 148955040300, 582714871830, 2281745337300, 8942420595810, 35074414899576, 137672461877850, 540756483094828
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OFFSET
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0,4
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COMMENTS
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A sawtooth pattern of length 1 is UD not followed by UD.
First differences of A024482.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
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FORMULA
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a(0)=0, a(1)=1, a(n) = A024482(n) - A024482(n-1) for n >= 2.
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MAPLE
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a:= proc(n) option remember; `if`(n<4, [0, 1, 1, 5][n+1],
(n-1)*(3*n-4)*(4*n-10)*a(n-1)/(n*(n-2)*(3*n-7)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Apr 24 2013
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MATHEMATICA
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Join[{0, 0, 1}, Table[(Binomial[2n, n]-Binomial[2n-2, n-1])/2, {n, 2, 25}]] // Differences (* Jean-François Alcover, Nov 12 2020 *)
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CROSSREFS
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Cf. A024482, A097613.
Sequence in context: A184309 A051944 A153373 * A166677 A318062 A158879
Adjacent sequences: A225012 A225013 A225014 * A225016 A225017 A225018
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KEYWORD
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nonn
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AUTHOR
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David Scambler, Apr 23 2013
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STATUS
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approved
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