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A277922 Number of positive walks with n steps {-4,-3,-2,-1,1,2,3,4} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

%I A277922 #12 Apr 03 2017 16:16:09
%S A277922 0,1,3,13,71,405,2501,15923,104825,704818,4827957,33549389,235990887,
%T A277922 1676907903,12019875907,86804930199,630999932585,4613307289260,
%U A277922 33900874009698,250257489686870,1854982039556397,13800559463237465,103017222722691145,771348369563479705
%N A277922 Number of positive walks with n steps {-4,-3,-2,-1,1,2,3,4} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.
%H A277922 Alois P. Heinz, <a href="/A277922/b277922.txt">Table of n, a(n) for n = 0..1000</a>
%H A277922 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.
%p A277922 b:= proc(n, y) option remember; `if`(n=0, `if`(y=1, 1, 0), add
%p A277922      ((h-> `if`(h<1, 0, b(n-1, h)))(y+d), d=[$-4..-1, $1..4]))
%p A277922     end:
%p A277922 a:= n-> b(n, 0):
%p A277922 seq(a(n), n=0..23);  # _Alois P. Heinz_, Nov 12 2016
%t A277922 b[n_, y_] := b[n, y] = If[n == 0, If[y == 1, 1, 0], Sum[Function[h, If[h < 1, 0, b[n - 1, h]]][y + d], {d, Join[Range[-4, -1], Range[4]]}]];
%t A277922 a[n_] := b[n, 0];
%t A277922 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Apr 03 2017, after _Alois P. Heinz_ *)
%Y A277922 Cf. A276852, A276901, A276902, A276903, A276904.
%K A277922 nonn,walk
%O A277922 0,3
%A A277922 _David Nguyen_, Nov 04 2016