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%I #25 Nov 04 2019 14:59:13
%S 1,0,1,1,2,2,6,6,17,21,54,74,183,272,644,1025,2342,3928,8734,15264,
%T 33227,59989,128484,238008,503563,952038,1995955,3835381,7987092,
%U 15548654,32223061,63388488,130918071,259724317,535168956,1069025128
%N Number of 3-generalized Motzkin paths of length n with no level steps H=(3,0) at odd level.
%H Robert Israel, <a href="/A257389/b257389.txt">Table of n, a(n) for n = 0..3087</a>
%F G.f.: (1-x^3-sqrt((1-x^3)*(1-4*x^2-x^3)))/(2*x^2*(1-x^3)).
%F a(n) = Sum_{k=0..n/3}(((-1)^(n-3*k)+1)*(binomial((n-k)/2,k)*(binomial(n-3*k,(n-3*k)/2))/((n-3*k+2)))). - _Vladimir Kruchinin_, Apr 02 2016
%F (2 + n)*a(n) + (14 + 4*n)*a(n + 1) + (-10 - 2*n)*a(n + 3) + (-20 - 4*n)*a(n + 4) + (8 + n)*a(n + 6) = 0. - _Robert Israel_, Nov 04 2019
%e For n=6 we have 6 paths: UDUDUD, H3H3, UUDUDD, UUUDDD, UDUUDD and UUDDUD, where H3=(3,0).
%p f:= gfun:-rectoproc({(2 + n)*a(n) + (14 + 4*n)*a(n + 1) + (-10 - 2*n)*a(n + 3) + (-20 - 4*n)*a(n + 4) + (8 + n)*a(n + 6), a(0) = 1, a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 2},a(n),remember):
%p map(f, [$0..100]); # _Robert Israel_, Nov 04 2019
%o (Maxima)
%o a(n):=sum(((-1)^(n-3*k)+1)*((binomial((n-k)/2,k) )*(binomial(n-3*k,(n-3*k)/2))/((n-3*k+2))),k,0,(n)/3); /* _Vladimir Kruchinin_, Apr 02 2016 */
%Y Cf. A000108, A090344, A086622, A257178, A007317, A064613, A104455, A104498.
%K nonn
%O 0,5
%A _José Luis Ramírez Ramírez_, Apr 21 2015