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 *)
A333504
Sum of the heights of all 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
0, 0, 1, 3, 9, 28, 88, 282, 921, 3058, 10302, 35159, 121406, 423704, 1493046, 5307276, 19014642, 68609686, 249149529, 910000728, 3341113126, 12325295866, 45664033813, 169846998495, 634020229888, 2374550269819, 8920273989351, 33604033638696, 126919824985533
Offset: 0
-
b:= proc(x, y, h) option remember; `if`(x=0, h, add((t->
`if`(x>t, b(x-1, t, max(h, t)), 0))(y-j), j=-1-y..min(1, y)))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..33);
-
b[x_, y_, h_] := b[x, y, h] = If[x == 0, h, Sum[With[{t = y - j},
If[x > t, b[x - 1, t, Max[h, t]], 0]], {j, -1 - y, Min[1, y]}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 33] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)
A333070
Total number of nodes summed over all 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, 2, 6, 16, 45, 132, 399, 1240, 3951, 12870, 42746, 144420, 495300, 1721202, 6051150, 21493136, 77039070, 278377452, 1013187920, 3711505380, 13675028346, 50649452084, 188482525039, 704409735912, 2642825539375, 9950643710800, 37587291143103, 142403408032648
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-> (n+1)*b(n, 0):
seq(a(n), n=0..30);
-
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_] := (n+1) b[n, 0];
a /@ Range[0, 30] (* 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 *)
Showing 1-4 of 4 results.