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%I #23 Jul 24 2022 13:09:17
%S 1,1,1,2,3,6,11,21,40,76,146,279,539,1036,2011,3883,7566,14662,28654,
%T 55692,109098,212564,417210,814568,1601366,3132078,6165732,12077905,
%U 23803827,46691096,92113651,180893091,357183430,702169718,1387539542,2730236900,5398831722,10632066436
%N Number of length n left factors of Dyck paths having no base pyramids.
%C 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)).
%H G. C. Greubel, <a href="/A191789/b191789.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = A191788(n,0).
%F G.f.: (1-z^2)*c/((1-z*c)*(1+z^4*c^2)), where c=(1-sqrt(1-4*z^2))/(2*z^2).
%F a(n) ~ 3*2^(n+1/2)/(5*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 21 2014
%F D-finite with recurrence -94*(n+1)*(n-15)*a(n) +2*(-47*n^2+734*n-2115)*a(n-1) +8*(88*n^2-1311*n+1307)*a(n-2) +2*(235*n^2-3904*n+13597)*a(n-3) +(-1593*n^2+25078*n-72033)*a(n-4) +(-423*n^2+7074*n-25079)*a(n-5) +(1241*n^2-19834*n+66805)*a(n-6) +2*(94*n^2-1585*n+5741)*a(n-7) -4*(n-7)*(117*n-1087)*a(n-8)=0. - _R. J. Mathar_, Jul 24 2022
%e a(4)=3 because we have UUDU, UUUD, and UUUU; each of the paths (UD)(UD), (UD)UU, and (UUDD) has at least one base pyramid (shown between parentheses).
%p c := ((1-sqrt(1-4*z^2))*1/2)/z^2: G := (1-z^2)*c/((1-z*c)*(1+z^4*c^2)): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, n), n = 0 .. 37);
%t With[{c=(1-Sqrt[1-4x^2])/(2x^2)},CoefficientList[Series[(1-x^2)c/ ((1-x c ) (1+x^4 c^2)), {x,0,40}],x]] (* _Harvey P. Dale_, Jun 19 2011 *)
%o (PARI) x='x+O('x^50); Vec( 2*(1-x^2)*(1-sqrt(1-4*x^2))/(x*(2*x-1+sqrt(1-4*x^2))*(3-2*x^2-sqrt(1-4*x^2))) ) \\ _G. C. Greubel_, Mar 27 2017
%Y Cf. A191788.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Jun 18 2011
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