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A175891 Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k)|0 which never go above the line y=x.

%I A175891 #17 Nov 11 2020 08:44:13
%S A175891 1,1,5,29,185,1226,8553,61642,455337,3429002,26229691,203237747,
%T A175891 1591820564,12582288455,100241042348,804090987555,6488942266564,
%U A175891 52644171729304,429123506792664,3512829202462126,28866426741057006,238031465396515626,1969001793889730276
%N A175891 Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k)|0<k<=4} which never go above the line y=x.
%H A175891 Alois P. Heinz, <a href="/A175891/b175891.txt">Table of n, a(n) for n = 0..1000</a>
%F A175891 a(n) ~ c * d^n / n^(3/2), where d = 8.84734830841870961487278801886633962039798... is the real root of the equation 4 + 4*d - 8*d^2 - 8*d^3 + d^4 = 0 and c = 0.31736815701423989167651891084531024477617724724822148387263881713... - _Vaclav Kotesovec_, May 30 2017
%p A175891 b:= proc(x, y) option remember; `if`(y>x or y<0, 0,
%p A175891       `if`(x=0, 1, add(b(x-j, y)+b(x, y-j), j=1..4)))
%p A175891     end:
%p A175891 a:= n-> b(n$2):
%p A175891 seq(a(n), n=0..35);  # _Alois P. Heinz_, May 16 2017
%t A175891 b[x_, y_] := b[x, y] = If[y > x || y < 0, 0, If[x == 0, 1, Sum[b[x - j, y] + b[x, y - j], {j, 1, 4}]]];
%t A175891 a[n_] := b[n, n];
%t A175891 a /@ Range[0, 35] (* _Jean-François Alcover_, Nov 11 2020, after _Alois P. Heinz_ *)
%Y A175891 Cf. A000103, A122951, A175883, A286918.
%K A175891 nonn
%O A175891 0,3
%A A175891 _Eric Werley_, Dec 05 2010