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 A186031 Number of Dyck paths of semilength n with a valley (DU) spanning the midpoint. 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 LINKS FORMULA a(n:even) = A000108(n-1); a(n:odd) = A000108(n-1) - A000108((n-1)/2)^2. EXAMPLE For n=4 the 5 paths are UDUD.UDUD, UUDD.UDUD, UDUD.UUDD, UUDD.UUDD and UUUD.UDDD where dot marks the midpoint. MAPLE 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); CROSSREFS 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 KEYWORD nonn AUTHOR David Scambler, Feb 11 2011 STATUS approved

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