|
%I #16 Feb 21 2024 08:27:54
%S 1,1,2,3,5,8,14,23,41,69,124,212,383,662,1200,2091,3799,6661,12122,
%T 21359,38919,68850,125578,222892,406865,724175,1322772,2360010,
%U 4313155,7711148,14099524,25252819,46192483,82863807,151628090,272385447,498578411,896774552
%N Number of length n left factors of Dyck paths having no UUDD's; here U=(1,1) and D=(1,-1).
%H Helmut Prodinger, <a href="https://arxiv.org/abs/2402.13026">Dispersed Dyck paths revisited</a>, arXiv:2402.13026 [math.CO], 2024.
%F a(n) = A191793(n,0).
%F G.f.: g(z) = 2/(1-2*z+z^4+sqrt(1-4*z^2+2*z^4+z^8)).
%F D-finite with recurrence (n+1)*a(n) +2*(-1)*a(n-1) +4*(-n+1)*a(n-2) +2*(n-3)*a(n-4) +6*a(n-5) +(n-7)*a(n-8)=0. - _R. J. Mathar_, Jul 22 2022
%e a(4)=5 because we have UDUU, UDUD, UUDU, UUUD, and UUUU, where U=(1,1) and D=(1,-1) (the path UUDD does not qualify).
%p g := 2/(1-2*z+z^4+sqrt(1-4*z^2+2*z^4+z^8)): gser := series(g, z = 0, 42): seq(coeff(gser, z, n), n = 0 .. 37);
%t CoefficientList[Series[2/(1-2x+x^4+Sqrt[1-4x^2+2x^4+x^8]), {x,0,40}], x] (* _Harvey P. Dale_, Jun 19 2011 *)
%Y Cf. A191793.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Jun 18 2011
|
