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A166287
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Number of peak plateaux in all Dyck paths of semilength n with no UUU's and no DDD's (U=(1,1), D=(1,-1)). A peak plateau is a run of consecutive peaks that is preceded by an upstep U and followed by a down step D; a peak consists of an upstep followed by a downstep.
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1
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0, 0, 0, 1, 3, 8, 21, 53, 133, 334, 839, 2112, 5329, 13475, 34143, 86674, 220400, 561309, 1431522, 3655480, 9345287, 23916622, 61267207, 157088278, 403103955, 1035192885, 2660312103, 6841157380, 17603254230, 45321606641, 116748360064
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| a(n)=Sum(k*A166285(n,k), k>=0).
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FORMULA
| G.f.: G=(1-z-z^2-h)/[2(1-z)h], where h = sqrt((1-3z+z^2)(1+z+z^2)).
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EXAMPLE
| a(4)=3 because we have UDUDUDUD, UDUDUUDD, UDUUDDUD, UD(UUDUDD), UUDDUDUD, UUDDUUDD, (UUDUDD)UD, (UUDUDUDD) (the 3 peak plateaux are shown between parentheses).
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MAPLE
| h := sqrt((1-3*z+z^2)*(1+z+z^2)): G := ((1-z-z^2-h)*1/2)/((1-z)*h): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);
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CROSSREFS
| A166285
Sequence in context: A014396 A170881 A039671 * A186812 A027930 A038200
Adjacent sequences: A166284 A166285 A166286 * A166288 A166289 A166290
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 12 2009
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