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

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Example

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).

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);

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 *)

Prog

(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

Crossrefs

Cf. A191525.

Sequence in context: A018184 A192675 A018185 * A005255 A086445 A127602

Adjacent sequences: A191523 A191524 A191525 * A191527 A191528 A191529

Keyword

nonn

Author

Emeric Deutsch, Jun 06 2011

Status

approved