A191520 - OEIS

%I #20 Dec 26 2023 09:58:12

%S 0,0,0,0,0,0,1,2,9,18,57,114,312,624,1578,3156,7599,15198,35401,70802,

%T 161052,322104,719790,1439580,3173090,6346180,13836426,27672852,

%U 59803104,119606208,256596276,513192552,1094249019,2188498038,4642178601,9284357202

%N Number of UUU's in all the dispersed Dyck paths of semilength n (i.e., in all Motzkin paths of length n; U=(1,1)).

%H G. C. Greubel, <a href="/A191520/b191520.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..floor((n-3)/2)} k*A191518(n,k) for n>=4 (clarified by _G. C. Greubel_).

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

%F a(n) ~ 2^(n-5/2)*sqrt(n)/sqrt(Pi) * (1 - 3*sqrt(Pi)/sqrt(2*n)). - _Vaclav Kotesovec_, Mar 21 2014

%F D-finite with recurrence n*(n-6)*a(n) -2*n*(n-6)*a(n-1) -4*(n-3)*(n-4)*a(n-2) +8*(n-3)*(n-4)*a(n-3)=0. - _R. J. Mathar_, Jul 24 2022

%e a(7)=2 because among the 35 (=A001405(7)) dispersed Dyck paths of length 7 only UUUDDDH and HUUUDDD have UUU's.

%p g := ((1-3*z^2-(1-z^2)*sqrt(1-4*z^2))*1/2)/((1-2*z)*sqrt(1-4*z^2)): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);

%t CoefficientList[Series[((1-3*x^2-(1-x^2)*Sqrt[1-4*x^2])*1/2)/((1-2*x)* Sqrt[1-4*x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)

%o (PARI) x='x+O('x^50); concat([0,0,0,0,0,0], Vec((1-3*x^2-(1-x^2)*sqrt(1-4*x^2))/(2*(1-2*x)*sqrt(1-4*x^2)))) \\ _G. C. Greubel_, Mar 26 2017

%Y Cf. A001405, A191518.

%K nonn

%O 0,8

%A _Emeric Deutsch_, Jun 07 2011