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A278395 Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-3,-2,-1,1,2,3}.

%I A278395 #26 Jul 01 2018 08:38:10
%S A278395 1,3,12,60,311,1674,9173,51002,286384,1620776,9228724,52810792,
%T A278395 303447096,1749612736,10117583749,58656027314,340806249367,
%U A278395 1984018271850,11569932938192,67574451148408,395214184047366,2314315680481252,13567587349336459,79621279809031310
%N A278395 Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-3,-2,-1,1,2,3}.
%C A278395 By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.
%H A278395 Andrew Howroyd, <a href="/A278395/b278395.txt">Table of n, a(n) for n = 0..200</a>
%H A278395 C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, <a href="https://arxiv.org/abs/1609.06473">Explicit formulas for enumeration of lattice paths: basketball and the kernel method</a>, arXiv:1609.06473 [math.CO], 2016.
%t A278395 frac[ex_] := Select[ex, Exponent[#, x] < 0&];
%t A278395 seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -3, 3}] - 1; p = 1/x; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
%t A278395 seq[24] (* _Jean-François Alcover_, Jul 01 2018, after _Andrew Howroyd_ *)
%o A278395 (PARI) seq(n)={my(v=vector(n), m=sum(i=-3, 3, x^i)-1, p=1/x); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p,x,1)); v} \\ _Andrew Howroyd_, Jun 27 2018
%Y A278395 Cf. A276852, A278391, A278392, A278393, A278394, A278396, A278398.
%K A278395 nonn,walk
%O A278395 0,2
%A A278395 _David Nguyen_, Nov 20 2016