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A186031
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Number of Dyck paths of semilength n with a valley (DU) spanning the midpoint.
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0
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0, 1, 1, 5, 10, 42, 107, 429, 1234, 4862, 15032, 58786, 190588, 742900, 2490399, 9694845, 33312770, 129644790, 453999656, 1767263190, 6282014804, 24466267020, 88026769844, 343059613650, 1246635155180, 4861946401452, 17815452662152
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OFFSET
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1,4
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LINKS
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Table of n, a(n) for n=1..27.
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FORMULA
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a(n:even) = A000108(n-1); a(n:odd) = A000108(n-1) - A000108((n-1)/2)^2.
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EXAMPLE
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For n=4 the 5 paths are UDUD.UDUD, UUDD.UDUD, UDUD.UUDD, UUDD.UUDD and UUUD.UDDD where dot marks the midpoint.
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MAPLE
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C:= n-> binomial (2*n, n)/(n+1):
a:= n-> C(n-1) -`if` (irem(n, 2)=1, C((n-1)/2)^2, 0):
seq (a(n), n=1..40);
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CROSSREFS
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Dyck n-paths with peak at midpoint is given by A000108(n-1); For UU (or DD) at midpoint see A027302.
Sequence in context: A001692 A038070 A136138 * A187877 A122173 A083515
Adjacent sequences: A186028 A186029 A186030 * A186032 A186033 A186034
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KEYWORD
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nonn
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AUTHOR
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David Scambler, Feb 11 2011
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STATUS
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approved
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