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A253831 Number of 2-Motzkin paths with no level steps at height 1.

%I A253831 #42 Feb 22 2018 22:24:03
%S A253831 1,2,5,12,30,76,197,522,1418,3956,11354,33554,102104,319608,1027237,
%T A253831 3381714,11371366,38946892,135505958,477781296,1703671604,6132978608,
%U A253831 22256615602,81327116484,298938112816,1104473254912,4098996843500,15272792557230,57106723430892,214202598271360,805743355591301
%N A253831 Number of 2-Motzkin paths with no level steps at height 1.
%C A253831 For n=3 we have 12 paths: H(1)H(1)H(1), H(1)H(1)H(2), H(1)H(2)H(1), H(1)H(2)H(2), H(2)H(1)H(1), H(2)H(1)H(2), H(2)H(2)H(1), H(2)H(2)H(2), UDH(1), UDH(2), H(1)UD, H(2)UD.
%H A253831 Robert Israel, <a href="/A253831/b253831.txt">Table of n, a(n) for n = 0..1000</a>
%F A253831 G.f.: 1/(1-2*x-x*F(x)), where F(x) is the g.f. of Fine numbers A000957.
%F A253831 G.f.: 2*(2+x)/(4-7*x-6*x^2+x*sqrt(1-4*x)).
%F A253831 a(n) ~ 4^(n+1) / (25*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 21 2015
%F A253831 (54+36*n)*a(n)+(-3+7*n)*a(n+1)+(-60-36*n)*a(n+2)+(36+16*n)*a(n+3)+(-6-2*n)*a(n+4) = 0. - _Robert Israel_, Apr 29 2015
%F A253831 a(n) = Sum_{m=1..n/2}(Sum_{j=0..n-2*m}(((Sum_{k=0..j}((k+1)*binomial(k+m,k+1)*binomial(2*j-k+m-1,j-k)*(-1)^(k)))*2^(n-j-2*m)*binomial(n-m-j,m))/(j+m)))+2^n. - _Vladimir Kruchinin_, Mar 11 2016
%p A253831 rec:= (54+36*n)*a(n)+(-3+7*n)*a(n+1)+(-60-36*n)*a(n+2)+(36+16*n)*a(n+3)+(-6-2*n)*a(n+4) = 0:
%p A253831 f:= gfun:-rectoproc({rec,seq(a(i)=[1,2,5,12][i+1],i=0..3)},a(n),remember):
%p A253831 seq(f(n),n=0..100); # _Robert Israel_, Apr 29 2015
%t A253831 CoefficientList[Series[1/(1-2*x-x*((1-Sqrt[1-4*x])/(3-Sqrt[1-4*x]))), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 21 2015 *)
%o A253831 (Maxima)
%o A253831 a(n):=sum(sum(((sum((k+1)*binomial(k+m,k+1)*binomial(2*j-k+m-1,j-k)*(-1)^(k),k,0,j))*2^(n-j-2*m)*binomial(n-m-j,m))/(j+m),j,0,n-2*m),m,1,n/2)+2^n; /* _Vladimir Kruchinin_, Mar 11 2016 */
%Y A253831 Cf. A000957, A217312, A257363.
%K A253831 nonn
%O A253831 0,2
%A A253831 _José Luis Ramírez Ramírez_, Apr 20 2015