%I #17 Sep 08 2022 08:45:57

%S 1,1,1,1,1,1,2,3,8,13,31,49,109,170,371,581,1270,2010,4417,7063,15581,

%T 25123,55554,90179,199752,326089,723351,1186670,2635764,4342829,

%U 9657336,15973459,35558165,59017088,131500422,218932442,488234057,815127111,1819186163

%N Number of dispersed Dyck paths of length n (i.e., of Motzkin paths of length n with no (1,0)-steps at positive heights) having no base pyramids. A base pyramid is a factor of the form U^j D^j (j>0), starting at the horizontal axis. Here U=(1,1) and D=(1,-1).

%C a(n) = A191392(n,0).

%H Vincenzo Librandi, <a href="/A191393/b191393.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(n) ~ 9 * 2^(n-11/2) * (16+(-1)^n) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014

%F D-finite with recurrence (n+1)*a(n) +3*(-n-1)*a(n-1) +(-5*n+13)*a(n-2) +(19*n-35)*a(n-3) +3*(n-11)*a(n-4) +2*(-17*n+73)*a(n-5) +(5*n-13)*a(n-6) +2*(13*n-77)*a(n-7) +4*(-n+8)*a(n-8) +8*(-n+8)*a(n-9)=0. - _R. J. Mathar_, Jul 26 2022

%e a(7)=3 because we have HHHHHHH, HUUDUDD, and UUDUDDH, where U=(1,1), D=(1,-1), and H=(1,0).

%p g := (2*(1-z^2))/(1-2*z+z^2+2*z^3+(1-z^2)*sqrt(1-4*z^2)): gser := series(g, z = 0, 42): seq(coeff(gser, z, n), n = 0 .. 38);

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

%o (Magma) m:=40; R<x>:=LaurentSeriesRing(RationalField(), m); Coefficients(R! (2*(1-x^2))/(1-2*x+x^2+2*x^3+(1-x^2)*Sqrt(1-4*x^2))); // _Vincenzo Librandi_, Mar 21 2014

%Y Cf. A191392.

%K nonn

%O 0,7

%A _Emeric Deutsch_, Jun 04 2011