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A278393 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 {-4,-3,-2,-1,0,1,2,3,4}.

%I A278393 #30 Jul 01 2018 08:38:00
%S A278393 1,4,26,194,1521,12289,101205,844711,7120398,60477913,516774114,
%T A278393 4437360897,38256405777,330948944639,2871299293535,24973776734091,
%U A278393 217690276938940,1901204163460913,16632544424086901,145730139895667887,1278596503973570665,11231908572986043199
%N A278393 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 {-4,-3,-2,-1,0,1,2,3,4}.
%C A278393 By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.
%H A278393 Andrew Howroyd, <a href="/A278393/b278393.txt">Table of n, a(n) for n = 0..200</a>
%H A278393 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 A278393 frac[ex_] := Select[ex, Exponent[#, x] < 0&];
%t A278393 seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -4, 4}]; 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 A278393 seq[22] (* _Jean-François Alcover_, Jul 01 2018, after _Andrew Howroyd_ *)
%o A278393 (PARI) seq(n)={my(v=vector(n), m=sum(i=-4, 4, x^i), 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 A278393 Cf. A276852, A278391, A278392, A278394, A278395, A278396, A278398.
%K A278393 nonn,walk
%O A278393 0,2
%A A278393 _David Nguyen_, Nov 20 2016