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A175883 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 A175883 #12 Nov 11 2020 08:53:29
%S A175883 1,1,5,29,170,1093,7346,50957,362476,2629150,19371533,144585146,
%T A175883 1090886362,8306621114,63752890716,492671044866,3830272606911,
%U A175883 29937476853483,235104315621495,1854181694878573,14679397763545597,116619744085592959,929412502842262520
%N A175883 Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k)|0<k<=3} which never go above the line y=x.
%H A175883 Alois P. Heinz, <a href="/A175883/b175883.txt">Table of n, a(n) for n = 0..1000</a>
%e A175883 a(3)=29 because we can reach (3,3) in the following ways:
%e A175883 by getting to (3,2) in 17 ways and then taking step (0,1), or
%e A175883 by getting to (3,1) in 8 ways and then taking step (0,2), or
%e A175883 by getting to (3,0) in 4 ways and then taking step (0,3).
%p A175883 b:= proc(x, y) option remember; `if`(y>x or y<0, 0,
%p A175883       `if`(x=0, 1, add(b(x-j, y)+b(x, y-j), j=1..3)))
%p A175883     end:
%p A175883 a:= n-> b(n$2):
%p A175883 seq(a(n), n=0..35);  # _Alois P. Heinz_, May 16 2017
%t A175883 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, 3}]]];
%t A175883 a[n_] := b[n, n];
%t A175883 a /@ Range[0, 35] (* _Jean-François Alcover_, Nov 11 2020, after _Alois P. Heinz_ *)
%Y A175883 Cf. A000108, A122951.
%K A175883 nonn
%O A175883 0,3
%A A175883 _Eric Werley_, Dec 05 2010