A257388 Number of 4-Motzkin paths of length n with no level steps at odd level.

1, 4, 17, 72, 306, 1304, 5573, 23888, 102702, 442904, 1915978, 8314480, 36195236, 158067312, 692475053, 3043191200, 13415404246, 59321085720, 263100680926, 1170347803440, 5221037429948, 23356788588752, 104772374565666, 471214329434208, 2124649562373708, 9603094073668208

Offset

0,2

Links

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

Formula

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

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

a(n) ~ sqrt(58+41*sqrt(2)) * 2^(n+1/2) * (1+sqrt(2))^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 22 2015

Conjecture: (n+2)*a(n) +8*(-n-1)*a(n-1) +4*(3*n+1)*a(n-2) +8*(2*n-3)*a(n-3)=0. - R. J. Mathar, Sep 24 2016

G.f. A(x) satisfies: A(x) = 1/(1 - 4*x) + x^2 * A(x)^2. - Ilya Gutkovskiy, Jun 30 2020

Example

For n=2 we have 17 paths: H(1)H(1), H(1)H(2), H(1)H(3), H(1)H(4), H(2)H(1), H(2)H(2), H(2)H(3), H(2)H(4), H(3)H(1), H(3)H(2), H(3)H(3), H(3)H(4), H(4)H(1), H(4)H(2), H(4)H(3), H(4)H(4) and UD.

Mathematica

CoefficientList[Series[(1-4*x-Sqrt[(1-4*x)*(1-4*x-4*x^2)])/(2*x^2*(1-4*x)), {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*(1-4*x))) \\ G. C. Greubel, Apr 08 2017

Crossrefs

Cf. A086622, A090344, A104498, A257178.

Sequence in context: A255117 A001076 A122451 * A113442 A362908 A085732

Adjacent sequences: A257385 A257386 A257387 * A257389 A257390 A257391

Keyword

nonn

Author

José Luis Ramírez Ramírez, Apr 21 2015

Status

approved