A191526 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.
1, 1, 1, 2, 4, 7, 13, 24, 46, 86, 166, 314, 610, 1163, 2269, 4352, 8518, 16414, 32206, 62292, 122464, 237590, 467842, 909960, 1794196, 3497248, 6903352, 13480826, 26635774, 52097267, 103020253, 201780224, 399300166, 783051638, 1550554582, 3044061116
Offset: 0
Keywords
Examples
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).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Crossrefs
Cf. A191525.
Programs
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Maple
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);
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Mathematica
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 *) -
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
Formula
a(n) = A191525(n,0).
G.f.: (((1+z)*sqrt(1-4*z^2)-(1-z)*(1-2*z))*1/2)/(z*(1-2*z)*(2+z^2)).
a(n) ~ 2^(n+3/2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 21 2014
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