A191786 Number of length n left factors of Dyck paths having no triple-rises (triple-rise = three consecutive (1,1)-steps).

1, 1, 2, 2, 4, 5, 9, 12, 22, 30, 55, 77, 141, 201, 368, 532, 974, 1424, 2607, 3847, 7043, 10474, 19176, 28707, 52559, 79133, 144888, 219234, 401420, 610073, 1117093, 1704380, 3120974, 4778408, 8750295, 13439431, 24611355, 37907920, 69422324, 107205933, 196336893

Offset

0,3

Comments

a(n)=A191785(n,0).

Links

Table of n, a(n) for n=0..40.

Formula

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

a(n) ~ 3^((n+3)/2) * (11+6*sqrt(3) + (11-6*sqrt(3))*(-1)^n) / (2*n^(3/2)* sqrt(2*Pi)). - Vaclav Kotesovec, Mar 21 2014

Conjecture: -(n+3)*(13*n-70)*a(n) +(-13*n^2+19*n-102)*a(n-1) +(65*n^2-221*n-516) *a(n-2) +(65*n^2-197*n+288)*a(n-3) -(n+6)*(13*n-97) *a(n-4) +3*(-13*n^2+35*n-70) *a(n-5) +(-169*n^2+1201*n-2208) *a(n-6) -9*(13*n-40)*(n-5) *a(n-7) -6*(13*n-25)*(n-6) *a(n-8)=0. - R. J. Mathar, Jun 14 2016

Example

a(4)=4 because we have UDUD, UDUU, UUDD, and UUDU, where U=(1,1), D=(1,-1); the paths UUUD and UUUU do not qualify.

Maple

g := (2*(1+z+z^2))/(1-z^2-2*z^3+sqrt(1-2*z^2-3*z^4)): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);

Mathematica

CoefficientList[Series[(2*(1+x+x^2))/(1-x^2-2*x^3+Sqrt[1-2*x^2-3*x^4]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 21 2014 *)

Crossrefs

Cf. A191785

Sequence in context: A001224 A102526 A050192 * A007147 A230380 A127968

Adjacent sequences: A191783 A191784 A191785 * A191787 A191788 A191789

Keyword

nonn

Author

Emeric Deutsch, Jun 18 2011

Status

approved