|
%I #25 Jun 30 2020 07:59:11
%S 1,4,17,72,306,1304,5573,23888,102702,442904,1915978,8314480,36195236,
%T 158067312,692475053,3043191200,13415404246,59321085720,263100680926,
%U 1170347803440,5221037429948,23356788588752,104772374565666,471214329434208,2124649562373708,9603094073668208
%N Number of 4-Motzkin paths of length n with no level steps at odd level.
%H G. C. Greubel, <a href="/A257388/b257388.txt">Table of n, a(n) for n = 0..1000</a>
%F 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.
%F G.f.: (1-4*x-sqrt((1-4*x)*(1-4*x-4*x^2)))/(2*x^2*(1-4*x)).
%F 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
%F 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
%F G.f. A(x) satisfies: A(x) = 1/(1 - 4*x) + x^2 * A(x)^2. - _Ilya Gutkovskiy_, Jun 30 2020
%e 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.
%t 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 *)
%o (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
%Y Cf. A086622, A090344, A104498, A257178.
%K nonn
%O 0,2
%A _José Luis Ramírez Ramírez_, Apr 21 2015
|
