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A306813 Number of 2n-step paths from (0,0) to (0,n) that stay in the first quadrant (but may touch the axes) consisting of steps (-1,0), (0,1), (0,-1) and (1,-1).

%I A306813 #28 May 13 2020 10:38:49
%S A306813 1,0,3,10,20,237,770,3944,28635,112360,744084,4381083,21579779,
%T A306813 143815322,801165187,4578481584,29176623983,165772480380,
%U A306813 1013147794546,6259309820475,36974951346176,230752749518819,1413352914731005,8618746801792237,53986291171211635
%N A306813 Number of 2n-step paths from (0,0) to (0,n) that stay in the first quadrant (but may touch the axes) consisting of steps (-1,0), (0,1), (0,-1) and (1,-1).
%H A306813 Vaclav Kotesovec, <a href="/A306813/b306813.txt">Table of n, a(n) for n = 0..1116</a> (terms 0..500 from Alois P. Heinz)
%F A306813 a(n) = A306814(2n,n).
%F A306813 a(n) ~ c * d^n / n^2, where d = 6.7004802541941947450873... and c = 0.5171899701803656646... - _Vaclav Kotesovec_, Apr 13 2019
%e A306813 a(0) = 1: [(0,0)].
%e A306813 a(2) = 3:
%e A306813   [(0,0), (0,1), (0,0), (0,1), (0,2)],
%e A306813   [(0,0), (0,1), (0,2), (0,1), (0,2)],
%e A306813   [(0,0), (0,1), (0,2), (0,3), (0,2)].
%e A306813 a(3) = 10:
%e A306813   [(0,0), (0,1), (1,0), (1,1), (1,2), (1,3), (0,3)],
%e A306813   [(0,0), (0,1), (0,2), (1,1), (1,2), (1,3), (0,3)],
%e A306813   [(0,0), (0,1), (0,2), (0,3), (1,2), (1,3), (0,3)],
%e A306813   [(0,0), (0,1), (0,2), (0,3), (0,4), (1,3), (0,3)],
%e A306813   [(0,0), (0,1), (1,0), (1,1), (1,2), (0,2), (0,3)],
%e A306813   [(0,0), (0,1), (0,2), (1,1), (1,2), (0,2), (0,3)],
%e A306813   [(0,0), (0,1), (0,2), (0,3), (1,2), (0,2), (0,3)],
%e A306813   [(0,0), (0,1), (1,0), (1,1), (0,1), (0,2), (0,3)],
%e A306813   [(0,0), (0,1), (0,2), (1,1), (0,1), (0,2), (0,3)],
%e A306813   [(0,0), (0,1), (1,0), (0,0), (0,1), (0,2), (0,3)].
%p A306813 b:= proc(n, x, y) option remember; `if`(min(n, x, y, n-x-y)<0, 0,
%p A306813       `if`(n=0, 1, add(b(n-1, x-d[1], y-d[2]),
%p A306813        d=[[-1, 0], [0, 1], [0, -1], [1, -1]])))
%p A306813     end:
%p A306813 a:= n-> b(2*n, 0, n):
%p A306813 seq(a(n), n=0..30);
%t A306813 b[n_, x_, y_] := b[n, x, y] = If[Min[n, x, y, n - x - y] < 0, 0, If[n == 0, 1, Sum[b[n - 1, x - d[[1]], y - d[[2]]], {d, {{-1, 0}, {0, 1}, {0, -1}, {1, -1}}}]]];
%t A306813 a[n_] := b[2n, 0, n];
%t A306813 a /@ Range[0, 30] (* _Jean-François Alcover_, May 13 2020, after Maple *)
%Y A306813 Cf. A000108, A126596, A174687, A306814, A317782.
%K A306813 nonn,walk
%O A306813 0,3
%A A306813 _Alois P. Heinz_, Mar 11 2019