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A257178 Number of 3-Motzkin paths of length n with no level steps at odd level. 4
1, 3, 10, 33, 110, 369, 1247, 4245, 14558, 50295, 175029, 613467, 2165100, 7692345, 27504600, 98941185, 357952580, 1301960925, 4759282415, 17478557925, 64468072820, 238736987535, 887359113700, 3309489922743, 12381998910700, 46460457776739 (list; graph; refs; listen; history; text; internal format)
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)}3^(n-2i)*C(i)*binomial(n-i,i), where C(n) is the n-th Catalan number A000108.

G.f.: (1-3*z-sqrt((1-3*z)*(1-3*z-4*z^2)))/(2*z^2*(1-3*z)).

a(n) ~ sqrt(5) * 4^(n+1) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015

Conjecture: (n+2)*a(n) +6*(-n-1)*a(n-1) +(5*n+4)*a(n-2) +6*(2*n-3)*a(n-3)=0. - R. J. Mathar, Sep 24 2016

EXAMPLE

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

MATHEMATICA

CoefficientList[Series[(1-3*x-Sqrt[(1-3*x)*(1-3*x-4x^2)])/(2*x^2*(1-3*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)

PROG

(PARI) Vec((1-3*x-sqrt((1-3*x)*(1-3*x-4*x^2)))/(2*x^2*(1-3*x)) + O(x^50)) \\ G. C. Greubel, Feb 05 2017

CROSSREFS

Cf. A090344, A086622.

Sequence in context: A289450 A113299 A126931 * A257363 A071722 A058987

Adjacent sequences:  A257175 A257176 A257177 * A257179 A257180 A257181

KEYWORD

nonn

AUTHOR

José Luis Ramírez Ramírez, Apr 20 2015

STATUS

approved

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Last modified June 6 01:16 EDT 2020. Contains 334858 sequences. (Running on oeis4.)