A166297 - OEIS

%I #21 Jul 24 2022 14:33:30

%S 0,0,0,1,2,5,12,28,66,156,370,882,2112,5079,12264,29725,72298,176414,

%T 431754,1059595,2607090,6429913,15893330,39365876,97692372,242875105,

%U 604836072,1508619585,3768496102,9426815859,23612178180,59217406914

%N 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)).

%C a(n) = Sum_{k=0..floor(n/3)} k*A166295(n,k).

%F G.f.: G(z) = 4*z^3/(1-z-z^2+sqrt(1-2*z-z^2-2*z^3+z^4))^2.

%F a(n) = A004148(n+1) - A004148(n) - A004148(n-1) for n>=3. - _Emeric Deutsch_, Nov 10 2009

%F a(n) ~ sqrt(5 + 3*sqrt(5)) * ((3+sqrt(5))/2)^n / (sqrt(2*Pi) * n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014

%F 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

%F 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

%e a(3)=1 because in UDUDUD, UDUUDD, UUDDUD, and UUDUDD we have 0+0+0+1=1 UUDUDD's starting at level 0.

%p 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);

%t 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 *)

%o (Maxima)

%o 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 */

%Y Cf. A166295.

%Y Cf. A004148. - _Emeric Deutsch_, Nov 10 2009

%K nonn

%O 0,5

%A _Emeric Deutsch_, Oct 29 2009