A257390 Number of 4-Motzkin paths of length n with no level steps at even level.

1, 0, 1, 4, 18, 80, 357, 1596, 7150, 32096, 144362, 650568, 2937316, 13286368, 60205805, 273290988, 1242639446, 5659468736, 25816338046, 117945079736, 539646216188, 2472638868960, 11345220210658, 52124831171544, 239792244636876, 1104495824173376

Offset: 0

Author

José Luis Ramírez Ramírez, Apr 21 2015

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..1322

Crossrefs

Cf. A090345, A025266, A257290.

Programs

Maple

rec:= a(n) = ((-16*n + 40)*a(n-3) + (-12*n+12)*a(n-2) +(8*n+4)*a(n-1))/(n+2):
f:= gfun:-rectoproc({rec,a(0)=1,a(1)=0,a(2)=1},a(n),remember):
seq(f(i),i=0..100); # Robert Israel, Apr 22 2015

Mathematica

CoefficientList[Series[(1-4*x-Sqrt[(1-4*x)*(1-4*x-4*x^2)])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 22 2015 *)

PARI

x='x+O('x^50); Vec((1-4*x-sqrt((1-4*x)*(1-4*x-4*x^2)))/(2*x^2)) \\ G. C. Greubel, Apr 08 2017

Formula

a(n) = Sum_{i=0..floor(n/2)}4^(n-2i)*C(i)*binomial(n-i-1,n), where C(i) is the i-th Catalan number A000108.

G.f.: (1-4*x-sqrt((1-4*x)*(1-4*x-4*x^2)))/(2*x^2).

a(n) ~ 2^(n+3/4) * (1+sqrt(2))^(n+1/2) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 22 2015

a(n) = ((-16*n + 40)*a(n-3) + (-12*n+12)*a(n-2) +(8*n+4)*a(n-1))/(n+2). - Robert Israel, Apr 22 2015