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

0,4

Links

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

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

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 *)

Prog

(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

Crossrefs

Cf. A090345, A025266, A257290.

Sequence in context: A037965 A045902 A090017 * A387401 A104631 A106391

Adjacent sequences: A257387 A257388 A257389 * A257391 A257392 A257393

Keyword

nonn

Author

José Luis Ramírez Ramírez, Apr 21 2015

Status

approved