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 A166297 Number of UUDUDD's starting at level 0 in all Dyck paths of semilength n with no UUU's and no DDD's (U=(1,1), D=(1,-1)). 2
 0, 0, 0, 1, 2, 5, 12, 28, 66, 156, 370, 882, 2112, 5079, 12264, 29725, 72298, 176414, 431754, 1059595, 2607090, 6429913, 15893330, 39365876, 97692372, 242875105, 604836072, 1508619585, 3768496102, 9426815859, 23612178180, 59217406914 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS a(n) = Sum_{k=0..floor(n/3)} k*A166295(n,k). LINKS FORMULA G.f.: G(z) = 4z^3/(1-z-z^2+sqrt(1-2*z-z^2-2*z^3+z^4))^2. a(n) = A004148(n+1) - A004148(n) - A004148(n-1) for n>=3. - Emeric Deutsch, Nov 10 2009 a(n) ~ sqrt(5 + 3*sqrt(5)) * ((3+sqrt(5))/2)^n / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 20 2014 EXAMPLE a(3)=1 because in UDUDUD, UDUUDD, UUDDUD, and UUDUDD we have 0+0+0+1=1 UUDUDD's starting at level 0. MAPLE G := 4*z^3/(1-z-z^2+sqrt(1-2*z-z^2-2*z^3+z^4))^2: Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32); MATHEMATICA CoefficientList[Series[4*x^3/(1-x-x^2+Sqrt[1-2*x-x^2-2*x^3+x^4])^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *) CROSSREFS Cf. A166295. Cf. A004148. - Emeric Deutsch, Nov 10 2009 Sequence in context: A302020 A239333 A255115 * A024960 A291234 A238828 Adjacent sequences:  A166294 A166295 A166296 * A166298 A166299 A166300 KEYWORD nonn AUTHOR Emeric Deutsch, Oct 29 2009 STATUS approved

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Last modified July 23 16:14 EDT 2019. Contains 325258 sequences. (Running on oeis4.)