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A256938 Expansion of g.f.: (1-3*z-sqrt(1-6*z+5*z^2+8*z^3-4*z^4))/(2*z^2*(1-z)).

%I A256938 #35 Jun 03 2017 19:01:28
%S A256938 1,2,7,24,86,316,1189,4562,17796,70398,281812,1139658,4649402,
%T A256938 19112962,79096155,329258424,1377798890,5792421108,24454224310,
%U A256938 103631241912,440674939192,1879769835968,8041447249926,34490981798188,148295899087660,639036278210420
%N A256938 Expansion of g.f.: (1-3*z-sqrt(1-6*z+5*z^2+8*z^3-4*z^4))/(2*z^2*(1-z)).
%C A256938 a(n) = number of lattice paths, never going below the x-axis, from (0,0) to (n,0) consisting of up steps U = (1,1), down steps D = (1,-1) and 2-colored horizontal steps H(k) = (k,0) for every positive integer k.
%H A256938 G. C. Greubel, <a href="/A256938/b256938.txt">Table of n, a(n) for n = 0..1000</a>
%H A256938 R. De Castro, A. L. Ramírez and J. L. Ramírez, <a href="http://dx.doi.org/10.7561/SACS.2014.1.137">Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs</a>, Scientific Annals of Computer Science, 24(1)(2014), 137-171
%F A256938 a(s) = Sum_{n=0..s} (Sum_{m=0..(s-2*n)} (C(n)*binomial(m+2*n,m) *binomial(s-2*n-1,m-1) * 2^m)), where C(n) = A000108(n).
%F A256938 G.f.: (1-3*z-sqrt(1-6*z+5*z^2+8*z^3-4*z^4))/(2*z^2*(1-z)).
%F A256938 a(n) ~ sqrt(221 + 53*sqrt(17)) * (5+sqrt(17))^n / (sqrt(Pi) * n^(3/2) * 2^(n+2)). - _Vaclav Kotesovec_, Apr 20 2015
%F A256938 Recurrence: (n+2)*a(n) = (7*n+5)*a(n-1) - (11*n-2)*a(n-2) - 3*(n-5)*a(n-3) + 12*(n-3)*a(n-4) - 4*(n-4)*a(n-5). - _Vaclav Kotesovec_, Apr 20 2015
%t A256938 CoefficientList[Series[(1-3*x-Sqrt[1-6*x+5*x^2+8*x^3-4*x^4])/(2*x^2*(1-x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 20 2015 *)
%o A256938 (PARI) x='x+O('x^50); Vec((1-3*x-sqrt(1 -6*x +5*x^2 +8*x^3 -4*x^4) )/(2*x^2*(1-x))) \\ _G. C. Greubel_, Jun 03 2017
%Y A256938 Cf. A135052.
%K A256938 nonn
%O A256938 0,2
%A A256938 _José Luis Ramírez Ramírez_, Apr 19 2015