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A278392 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,-1,0,1,2,3}.

%I A278392 #28 Jul 01 2018 08:37:55
%S A278392 1,3,15,87,530,3329,21316,138345,906853,5989967,39804817,265812731,
%T A278392 1782288408,11991201709,80911836411,547334588037,3710610424765,
%U A278392 25204313298581,171492983631249,1168638213247713,7974592724571446,54484621312318007,372671912259214487
%N A278392 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,-1,0,1,2,3}.
%C A278392 By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.
%H A278392 Andrew Howroyd, <a href="/A278392/b278392.txt">Table of n, a(n) for n = 0..200</a>
%H A278392 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 A278392 frac[ex_] := Select[ex, Exponent[#, x] < 0&];
%t A278392 seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -3, 3}]; 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 A278392 seq[23] (* _Jean-François Alcover_, Jul 01 2018, after _Andrew Howroyd_ *)
%o A278392 (PARI) seq(n)={my(v=vector(n), m=sum(i=-3, 3, 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 A278392 Cf. A276852, A278391, A278393, A278394, A278395, A278396, A278398.
%K A278392 nonn,walk
%O A278392 0,2
%A A278392 _David Nguyen_, Nov 20 2016