A333069
Number of lattice paths from (0,0) to (n,0) that do not go below the x-axis, and at (x,y) only allow steps (1,v) with v in {-1,0,1,...,y+1}.
Original entry on oeis.org
1, 1, 2, 4, 9, 22, 57, 155, 439, 1287, 3886, 12035, 38100, 122943, 403410, 1343321, 4531710, 15465414, 53325680, 185575269, 651191826, 2302247822, 8194892393, 29350405663, 105713021575, 382717065800, 1392121894189, 5085836001166, 18654616951435, 68678029247822
Offset: 0
-
b:= proc(x, y) option remember; `if`(x=0, 1, add(
`if`(x+j>y, b(x-1, y-j), 0), j=-1-y..min(1, y)))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..33);
-
b[x_, y_] := b[x, y] = If[x == 0, 1, Sum[If[x + j > y, b[x - 1, y - j], 0], {j, -1 - y, Min[1, y]}]];
a[n_] := b[n, 0];
a /@ Range[0, 33] (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)
A333608
Sum of the heights of all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (1,v) with v in {-1,0,...,max(y,1)}.
Original entry on oeis.org
0, 0, 1, 3, 9, 25, 70, 200, 584, 1742, 5304, 16471, 52120, 167885, 549856, 1828897, 6170108, 21087458, 72923515, 254880303, 899454849, 3201729220, 11486266036, 41497996004, 150879471934, 551723923040, 2027990653855, 7489507917594, 27777837416779, 103427750936183
Offset: 0
-
b:= proc(x, y, h) option remember;
`if`(x=0, h, add(b(x-1, y+j, max(y, h)),
j=-min(1, y)..min(max(1, y), x-y-1)))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..29);
-
b[x_, y_, h_] := b[x, y, h] = If[x == 0, h, Sum[b[x - 1, y + j, Max[y, h]], {j, -Min[1, y], Min[Max[1, y], x - y - 1]}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 29] (* Jean-François Alcover, May 12 2020, after Maple *)
A333106
Total number of nodes summed over all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (1,v) with v in {-1,0,...,max(y,1)}.
Original entry on oeis.org
1, 2, 6, 16, 45, 126, 357, 1024, 2979, 8800, 26422, 80688, 250705, 792568, 2548620, 8331568, 27667109, 93241152, 318569656, 1102246040, 3857916552, 13644697000, 48716177272, 175417870080, 636493447625, 2325399611652, 8548381939932, 31599848465276
Offset: 0
-
b:= proc(x, y) option remember; `if`(x=0, 1, add(
b(x-1, y+j), j=-min(1, y)..min(max(1, y), x-y-1)))
end:
a:= n-> (n+1)*b(n, 0):
seq(a(n), n=0..29);
-
b[x_, y_] := b[x, y] = If[x == 0, 1,
Sum[b[x-1, y+j], {j, -Min[1, y], Min[Max[1, y], x-y-1]}]];
a[n_] := (n+1) b[n, 0];
a /@ Range[0, 29] (* Jean-François Alcover, Apr 05 2021, after Alois P. Heinz *)
A333107
Total area under all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (1,v) with v in {-1,0,...,max(y,1)}.
Original entry on oeis.org
0, 0, 1, 4, 16, 56, 190, 637, 2131, 7156, 24215, 82758, 285991, 999715, 3534394, 12631420, 45601759, 166169360, 610650687, 2261234467, 8430749631, 31625520000, 119281312293, 452077280484, 1720796968459, 6575385383602, 25212139233077, 96970372087853
Offset: 0
-
b:= proc(x, y) option remember; `if`(x=0, [1, 0], add(
(p-> p+[0, p[1]*(y+j/2)])(b(x-1, y+j)),
j=-min(1, y)..min(max(1, y), x-y-1)))
end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..29);
-
b[x_, y_] := b[x, y] = If[x == 0, {1, 0}, Sum[
Function[p, p + {0, p[[1]]*(y + j/2)}][b[x - 1, y + j]],
{j, -Min[1, y], Min[Max[1, y], x - y - 1]}]];
a[n_] := b[n, 0][[2]];
a /@ Range[0, 29] (* Jean-François Alcover, Apr 05 2021, after Alois P. Heinz *)
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}.
Original entry on oeis.org
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
-
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 *)
A348202
Number of nonnegative lattice paths from (0,0) to (n,0) using steps in {(1,-4), (1,-1), (1,0), (1,1)}.
Original entry on oeis.org
1, 1, 2, 4, 9, 22, 57, 155, 435, 1249, 3645, 10770, 32143, 96747, 293359, 895373, 2748803, 8483035, 26302248, 81896176, 255967640, 802790415, 2525691721, 7968972542, 25209580699, 79942927651, 254077293876, 809192984902, 2582113984084, 8254273128869
Offset: 0
-
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, 1, add(b(x-1, y-j), j=[-4, -1, 0, 1])))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..31);
-
b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - 1, y - j], {j, {-4, -1, 0, 1}}]]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz *)
Showing 1-6 of 6 results.
Comments