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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) = 4*z^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
a(n) = 2*Sum_{k=0..(n-2)/2} C(n-k-1,k)*C(n-k-1,k+2)/(n-k-1). - Vladimir Kruchinin, Oct 13 2020
D-finite with recurrence +(n+3)*a(n) +(-3*n-4)*a(n-1) +(n-4)*a(n-3) +2*(2*n-7)*a(n-4) +(n-6)*a(n-5) +(-n+7)*a(n-6)=0. - R. J. Mathar, Jul 24 2022
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 *)
PROG
(Maxima)
a(n):=2*sum((binomial(n-1-k, k)*binomial(n-1-k, k+2))/(n-1-k), k, 0, (n-2)/2); /* Vladimir Kruchinin, Oct 13 2020 */
CROSSREFS
Cf. A166295.
Cf. A004148. - Emeric Deutsch, Nov 10 2009
Sequence in context: A302020 A239333 A255115 * A024960 A291234 A346051
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 29 2009
STATUS
approved

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)