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A348202 Number of nonnegative lattice paths from (0,0) to (n,0) using steps in {(1,-4), (1,-1), (1,0), (1,1)}.

%I A348202 #23 Dec 28 2022 05:42:55
%S A348202 1,1,2,4,9,22,57,155,435,1249,3645,10770,32143,96747,293359,895373,
%T A348202 2748803,8483035,26302248,81896176,255967640,802790415,2525691721,
%U A348202 7968972542,25209580699,79942927651,254077293876,809192984902,2582113984084,8254273128869
%N A348202 Number of nonnegative lattice paths from (0,0) to (n,0) using steps in {(1,-4), (1,-1), (1,0), (1,1)}.
%H A348202 Alois P. Heinz, <a href="/A348202/b348202.txt">Table of n, a(n) for n = 0..1907</a>
%H A348202 Alois P. Heinz, <a href="/A348202/a348202.gif">Animation of a(7) = 155 paths</a>
%H A348202 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A348202 a(n) ~ c * d^n / n^(3/2), where d = 3.3640233336410979391691803264403704977... is the root of the equation 256*d^5 - 1280*d^4 + 960*d^3 + 2267*d^2 - 1324*d - 4112 = 0 and c = 0.710307351107763693658610320440791667652705027171696102847138... - _Vaclav Kotesovec_, Oct 24 2021
%p A348202 b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
%p A348202      `if`(x=0, 1, add(b(x-1, y-j), j=[-4, -1, 0, 1])))
%p A348202     end:
%p A348202 a:= n-> b(n, 0):
%p A348202 seq(a(n), n=0..31);
%t A348202 b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - 1, y - j], {j, {-4, -1, 0, 1}}]]];
%t A348202 a[n_] := b[n, 0];
%t A348202 Table[a[n], {n, 0, 31}] (* _Jean-François Alcover_, Dec 28 2022, after _Alois P. Heinz_ *)
%Y A348202 Cf. A000108, A001006, A025235, A036765, A333069, A333105, A337067.
%K A348202 nonn,walk
%O A348202 0,3
%A A348202 _Alois P. Heinz_, Oct 06 2021