A257363 Number of 3-Motzkin paths with no level steps at height 1

1, 3, 10, 33, 110, 369, 1247, 4248, 14603, 50724, 178314, 635526, 2300829, 8477382, 31842897, 122103276, 478372886, 1915188093, 7831613468, 32674683984, 138871668314, 600140517762, 2631926843602, 11690520554421, 52498671870181, 237966449687118, 1087246253873875, 5001141997115010, 23137102115963262

Offset

0,2

Comments

For n=2 we have 10 paths: H(1)H(1), H(1)H(2), H(2)H(1), H(2)H(2), H(1)H(3), H(3)H(1), H(3)H(3), H(2)H(3), H(3)H(2), UD.

Links

Robert Israel, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1/(1-3*x-x*F(x)), where F(x) is the g.f. of the sequence A117641.

G.f.: 2*(3+x)/(6-17*x-9*x^2+x*sqrt(1-6*x+5*x^2)).

a(n) ~ 5^(n+3/2)/(98*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015

From Robert Israel, Apr 28 2015 (Start):

G.f.: (6-x*sqrt(1-6*x+5*x^2)-17*x-9*x^2)/(6-36*x+42*x^2+38*x^3).

3*(-n+1)*a(n) +9*(4*n-7)*a(n-1) +9*(-16*n+39)*a(n-2) +(197*n-656)*a(n-3) +9*(n+15)*a(n-4) +95*(-n+4)*a(n-5)=0. (End)

Maple

rec:= (95+95*n)*a(n)+(-180-9*n)*a(n+1)+(-329-197*n)*a(n+2)+(369+144*n)*a(n+3)+(-117-36*n)*a(4+n)+(12+3*n)*a(n+5):
f:= gfun:-rectoproc({rec, a(0)=1, a(1)=3, a(2)=10, a(3)=33, a(4)=110}, a(n), remember):
seq(f(n), n=0..100); # Robert Israel, Apr 28 2015

Mathematica

CoefficientList[Series[2*(3+x)/(6-17*x-9*x^2+x*Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)

Crossrefs

Cf. A117641, A217312, A253831, A059231.

Sequence in context: A113299 A126931 A257178 * A071722 A058987 A001558

Adjacent sequences: A257360 A257361 A257362 * A257364 A257365 A257366

Keyword

nonn

Author

José Luis Ramírez Ramírez, Apr 20 2015

Status

approved