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 A256939 Expansion of g.f.: (1-4*z-sqrt(1-8*z+12*z^2+8*z^3-4*z^4))/(2*z^2(1-z)). 0
 1, 3, 13, 57, 257, 1185, 5573, 26661, 129437, 636429, 3163725, 15877101, 80340813, 409495053, 2100558429, 10836262173, 56184433661, 292628726205, 1530338756093, 8032671187581, 42304703640701, 223484135199357, 1183921500416509, 6288098247289341 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 3-colored horizontal steps H(k) = (k,0) for every positive integer k. LINKS R. De Castro, A. L. Ramírez and J. L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, Scientific Annals of Computer Science, 24(1)(2014), 137-171. FORMULA a(s) = Sum_{n=0..s} (Sum_{m=0..s-2n} (C(n)binomial(m+2n,m)*binomial(s-2n-1,m-1)3^m)), where C(n)=A000108(n). G.f.: (1-4z-sqrt(1-8z+12z^2+8z^3-4z^4))/(2z^2(1-z)). a(n) ~ sqrt(77 + 29*sqrt(7)) * (3+sqrt(7))^n / (sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Apr 20 2015 Recurrence: (n+2)*a(n) = 3*(3*n+2)*a(n-1) - 4*(5*n-2)*a(n-2) + 4*(n+2)*a(n-3) + 12*(n-3)*a(n-4) - 4*(n-4)*a(n-5). - Vaclav Kotesovec, Apr 20 2015 MATHEMATICA CoefficientList[Series[(1-4*x-Sqrt[1-8*x+12*x^2+8*x^3-4*x^4])/(2*x^2*(1-x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2015 *) CROSSREFS Cf. A135052. Sequence in context: A275634 A163606 A115968 * A005827 A151319 A151222 Adjacent sequences:  A256936 A256937 A256938 * A256940 A256941 A256942 KEYWORD nonn AUTHOR José Luis Ramírez Ramírez, Apr 19 2015 STATUS approved

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Last modified May 29 10:05 EDT 2020. Contains 334699 sequences. (Running on oeis4.)