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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).
3
1, 0, 3, 10, 20, 237, 770, 3944, 28635, 112360, 744084, 4381083, 21579779, 143815322, 801165187, 4578481584, 29176623983, 165772480380, 1013147794546, 6259309820475, 36974951346176, 230752749518819, 1413352914731005, 8618746801792237, 53986291171211635
OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1116 (terms 0..500 from Alois P. Heinz)
FORMULA
a(n) = A306814(2n,n).
a(n) ~ c * d^n / n^2, where d = 6.7004802541941947450873... and c = 0.5171899701803656646... - Vaclav Kotesovec, Apr 13 2019
EXAMPLE
a(0) = 1: [(0,0)].
a(2) = 3:
[(0,0), (0,1), (0,0), (0,1), (0,2)],
[(0,0), (0,1), (0,2), (0,1), (0,2)],
[(0,0), (0,1), (0,2), (0,3), (0,2)].
a(3) = 10:
[(0,0), (0,1), (1,0), (1,1), (1,2), (1,3), (0,3)],
[(0,0), (0,1), (0,2), (1,1), (1,2), (1,3), (0,3)],
[(0,0), (0,1), (0,2), (0,3), (1,2), (1,3), (0,3)],
[(0,0), (0,1), (0,2), (0,3), (0,4), (1,3), (0,3)],
[(0,0), (0,1), (1,0), (1,1), (1,2), (0,2), (0,3)],
[(0,0), (0,1), (0,2), (1,1), (1,2), (0,2), (0,3)],
[(0,0), (0,1), (0,2), (0,3), (1,2), (0,2), (0,3)],
[(0,0), (0,1), (1,0), (1,1), (0,1), (0,2), (0,3)],
[(0,0), (0,1), (0,2), (1,1), (0,1), (0,2), (0,3)],
[(0,0), (0,1), (1,0), (0,0), (0,1), (0,2), (0,3)].
MAPLE
b:= proc(n, x, y) option remember; `if`(min(n, x, y, n-x-y)<0, 0,
`if`(n=0, 1, add(b(n-1, x-d[1], y-d[2]),
d=[[-1, 0], [0, 1], [0, -1], [1, -1]])))
end:
a:= n-> b(2*n, 0, n):
seq(a(n), n=0..30);
MATHEMATICA
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}}}]]];
a[n_] := b[2n, 0, n];
a /@ Range[0, 30] (* Jean-François Alcover, May 13 2020, after Maple *)
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Alois P. Heinz, Mar 11 2019
STATUS
approved