A257178 - OEIS

%I #23 Jun 30 2020 07:59:05

%S 1,3,10,33,110,369,1247,4245,14558,50295,175029,613467,2165100,

%T 7692345,27504600,98941185,357952580,1301960925,4759282415,

%U 17478557925,64468072820,238736987535,887359113700,3309489922743,12381998910700,46460457776739

%N Number of 3-Motzkin paths of length n with no level steps at odd level.

%H G. C. Greubel, <a href="/A257178/b257178.txt">Table of n, a(n) for n = 0..1000</a>

%F 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.

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

%F a(n) ~ sqrt(5) * 4^(n+1) / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 21 2015

%F 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

%F G.f. A(x) satisfies: A(x) = 1/(1 - 3*x) + x^2 * A(x)^2. - _Ilya Gutkovskiy_, Jun 30 2020

%e 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.

%t 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 *)

%o (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

%Y Cf. A090344, A086622.

%K nonn

%O 0,2

%A _José Luis Ramírez Ramírez_, Apr 20 2015