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A278416 Number of meanders (walks starting at the origin and ending at any altitude >= 0 that may touch but never go below the x-axis) with n steps from {-4,-3,-2,-1,1,2,3,4}.

%I A278416 #18 Jul 11 2018 08:36:07
%S A278416 1,4,26,174,1231,8899,65492,487646,3664123,27723979,210946444,
%T A278416 1612394958,12371547879,95230159650,735060394986,5687343753535,
%U A278416 44096482961189,342530654187820,2665058975987628,20765913987073659,162019898098364055,1265622208055843635
%N A278416 Number of meanders (walks starting at the origin and ending at any altitude >= 0 that may touch but never go below the x-axis) with n steps from {-4,-3,-2,-1,1,2,3,4}.
%H A278416 Andrew Howroyd, <a href="/A278416/b278416.txt">Table of n, a(n) for n = 0..200</a>
%H A278416 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 A278416 seq[n_] := Module[{v = Table[1, n], m = Sum[ x^i, {i, -4, 4}] - 1, p = 1}, For[i = 2, i <= n, i++, p = Expand[p*m]; p = p - Select[p, Exponent[#, x] < 0&]; v[[i]] = (p /. x -> 1)]; v];
%t A278416 seq[25] (* _Jean-François Alcover_, Jul 11 2018, after _Andrew Howroyd_ *)
%o A278416 (PARI) seq(n)={my(v=vector(n), m=sum(i=-4, 4, x^i)-1, p=1); 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 A278416 Cf. A001405, A047002, A278398, A278396, A205337.
%K A278416 nonn,walk
%O A278416 0,2
%A A278416 _Michael Wallner_, Nov 21 2016