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%I #12 Mar 29 2017 20:25:33
%S 1,1,1,2,4,7,13,24,46,86,166,314,610,1163,2269,4352,8518,16414,32206,
%T 62292,122464,237590,467842,909960,1794196,3497248,6903352,13480826,
%U 26635774,52097267,103020253,201780224,399300166,783051638,1550554582,3044061116
%N Number left factors of Dyck paths of length n and having no hills; a hill is a (1,1)-step starting at level 0 and followed by a (1,-1)-step.
%H G. C. Greubel, <a href="/A191526/b191526.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = A191525(n,0).
%F G.f.: (((1+z)*sqrt(1-4*z^2)-(1-z)*(1-2*z))*1/2)/(z*(1-2*z)*(2+z^2)).
%F a(n) ~ 2^(n+3/2)/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 21 2014
%F Conjecture: -2*(n+1)*(3*n-10)*a(n) +12*(n-5)*a(n-1) +(21*n^2-97*n+122)*a(n-2) +6*(n-5)*a(n-3) +4*(n-2)*(3*n-7)*a(n-4)=0. - _R. J. Mathar_, Jun 14 2016
%e a(4)=4 because the paths UUDD, UUDU, UUUD, and UUUU have no hills; here U=(1,1) and D=(1,-1) (UDUD and UDUU have 2 and 1 hills, respectively.
%p g := (((1+z)*sqrt(1-4*z^2)-(1-z)*(1-2*z))*1/2)/(z*(1-2*z)*(2+z^2)): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);
%t CoefficientList[Series[(((1+x)*Sqrt[1-4*x^2]-(1-x)*(1-2*x))*1/2)/(x*(1-2*x) *(2+x^2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)
%o (PARI) z='z+O('z^50); Vec((((1+z)*sqrt(1-4*z^2)-(1-z)*(1-2*z))*1/2)/(z*(1-2*z)*(2+z^2))) \\ _G. C. Greubel_, Mar 27 2017
%Y Cf. A191525.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Jun 06 2011
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