A337318 -id:A337318 - cp's OEIS frontend

A337067 Number of nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (h,v) with h in {1..max(1,y)} and v in {-1,0,1}.

1, 1, 2, 4, 9, 22, 57, 156, 447, 1332, 4103, 12999, 42176, 139638, 470353, 1607861, 5566543, 19484810, 68859862, 245404650, 881081082, 3184214751, 11575346316, 42300703150, 155316289004, 572725968326, 2120154235114, 7876449597257, 29356608044002

Offset: 0

Alois P. Heinz, Oct 12 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1671
Wikipedia, Counting lattice paths

Cf. A333069, A333105, A333647, A337318, A337863.

b:= proc(x, y) option remember; `if`(x=0, 1, add(add(
      b(x-h, y-v), h=1..min(x-y+v, max(1, y-v))), v=-1..min(y, 1)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);

b[x_, y_] := b[x, y] = If[x == 0, 1, Sum[Sum[
    b[x-h, y-v], {h, 1, Min[x-y+v, Max[1, y-v]]}], {v, -1, Min[y, 1]}]];
a[n_] := b[n, 0];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 22 2020, after Alois P. Heinz *)

a(n) ~ c * 4^n / n^(3/2), where c = 0.03828240225265266504281697555169550706277641504396262520878537702016362... - Vaclav Kotesovec, Oct 24 2020

A337863 Twice the total area of all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (h,v) with h in {1..max(1,y)} and v in {-1,0,1}.

Original entry on oeis.org

0, 0, 2, 8, 32, 123, 459, 1715, 6432, 24256, 91993, 350817, 1344559, 5176106, 20003683, 77567894, 301667089, 1176204195, 4596267116, 17995972170, 70581437130, 277244854010, 1090484851145, 4294324521033, 16929104620229, 66801975748076, 263827090563129

Offset: 0

Alois P. Heinz, Oct 12 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1663

Cf. A337067, A337318.

b:= proc(x, y) option remember; `if`(x=0, [1, 0],
      add(add((p-> p+[0, p[1]*h*(2*y-v)])(b(x-h, y-v)),
      h=1..min(x-y+v, max(1, y-v))), v=-1..min(y, 1)))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);

b[x_, y_] := b[x, y] = If[x == 0, {1, 0},
     Sum[Sum[Function[p, p + {0, p[[1]]*h*(2*y - v)}][b[x - h, y - v]],
     {h, 1, Min[x - y + v, Max[1, y - v]]}], {v, -1, Min[y, 1]}]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 24 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A337067 Number of nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (h,v) with h in {1..max(1,y)} and v in {-1,0,1}.

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