A191317 - OEIS

%I #20 Jul 22 2022 11:46:11

%S 1,1,2,3,5,8,14,23,40,67,117,198,346,590,1032,1769,3096,5328,9329,

%T 16103,28205,48801,85500,148216,259733,450952,790387,1374044,2408653,

%U 4191814,7349019,12801243,22445281,39127766,68611494,119687036,209890344,366348367,642493426,1121992447,1967839835

%N Number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) having no UDU's, where U=(1,1) and D=(1,-1).

%C a(n) = A191316(n,0).

%C Addendum Jun 18 2011: (Start)

%C Also the number of length n left factors of Dyck paths having no DUD's.

%C Also number of dispersed Dyck paths with no DUD's. Example: a(4)=5 because we have UDHH, UUDD, HUDH, HHUD, and HHHH (here H = (1,0)). (End)

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

%F D-finite with recurrence (n+1)*a(n) +2*(-n-1)*a(n-1) +(-n+5)*a(n-2) +3*(n-3)*a(n-3) +(-5*n+19)*a(n-4) +2*(4*n-17)*a(n-5) +3*(-n+5)*a(n-6) +3*(n-5)*a(n-7)=0. - _R. J. Mathar_, Jul 22 2022

%e a(4)=5 because we have HHHH, HHUD, HUDH, UDHH, and UUDD, where U=(1,1), D=(1,-1), and H=(1,0). (UDUD does not qualify.)

%e a(4)=5 because we have UDUU, UUDD, UUDU, UUUD, and UUUU (UDUD does not qualify).

%p g := ((sqrt(1-2*z^2-3*z^4)-1+2*z-z^2+2*z^3)*1/2)/(z*(1-2*z+z^2-z^3)): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);

%p # alternative, Jun 18 2011:

%p g := (2*(1+z^2))/((1-2*z)*(1+z^2)+sqrt((1+z^2)*(1-3*z^2))): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);

%Y Cf. A191316.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Jun 01 2011