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A278398 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 {-3,-2,-1,1,2,3}.

%I A278398 #26 Jul 01 2018 08:38:22
%S A278398 1,3,15,75,400,2169,11989,66985,377718,2144290,12240943,70193305,
%T A278398 404029950,2332989921,13508237399,78399357623,455959701700,
%U A278398 2656652705422,15504203678738,90614205677898,530288460288008,3107012752773125,18223934202102463,106996319699099591
%N A278398 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 {-3,-2,-1,1,2,3}.
%H A278398 Andrew Howroyd, <a href="/A278398/b278398.txt">Table of n, a(n) for n = 0..200</a>
%H A278398 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 A278398 frac[ex_] := Select[ex, Exponent[#, x] < 0&];
%t A278398 seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -3, 3}] - 1; p = 1; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
%t A278398 seq[24] (* _Jean-François Alcover_, Jul 01 2018, after _Andrew Howroyd_ *)
%o A278398 (PARI) seq(n)={my(v=vector(n), m=sum(i=-3, 3, 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 A278398 Cf. A276902, A276852, A278391, A278394.
%K A278398 nonn,walk
%O A278398 0,2
%A A278398 _David Nguyen_, Nov 20 2016