A257519 Number of Motzkin paths of length n with no peaks at level 4.

1, 1, 2, 4, 9, 21, 51, 127, 322, 827, 2145, 5607, 14751, 39020, 103713, 276848, 741901, 1995340, 5384554, 14576673, 39579527, 107776557, 294283193, 805649528, 2211176173, 6083560542, 16776970140, 46372110274, 128456563024, 356600559820, 991986172469, 2765030171165, 7722156349298, 21607098380159

Offset

0,3

Links

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

Formula

G.f.: 1/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x+x^2*(1-M(x)))))), where M(x) is the g.f. of Motzkin numbers A001006.

a(n) ~ 3^(n+7/2)/(98*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 27 2015

Example

For n=4 we have 9 paths: HHHH, UDUD, UHDH, HUHD, UHHD, UDHH, HUDH, HHUD and UUDD

Mathematica

CoefficientList[Series[1/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x+x^2*(1-(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2)))))), {x, 0, 30}], x] (* Vaclav Kotesovec, Apr 27 2015 *)

Prog

(PARI) x='x+O('x^50); Vec(1/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x+x^2*(1-(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2))))))) \\ G. C. Greubel, Jun 03 2017

Crossrefs

Cf. A089372, A257300, A257104.

Sequence in context: A051529 A230554 A005207 * A257387 A094286 A094287

Adjacent sequences: A257516 A257517 A257518 * A257520 A257521 A257522

Keyword

nonn

Author

José Luis Ramírez Ramírez, Apr 27 2015

Status

approved