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

Author

José Luis Ramírez Ramírez, Apr 20 2015

Keywords

• nonn

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

Crossrefs

Cf. A117641, A217312, A253831, A059231.

Programs

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 *)

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)