A333105
Number of 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, 1, 2, 4, 9, 21, 51, 128, 331, 880, 2402, 6724, 19285, 56612, 169908, 520723, 1627477, 5180064, 16766824, 55112302, 183710312, 620213500, 2118094664, 7309077920, 25459737905, 89438446602, 316606738516, 1128566016617, 4048230694964, 14604517154115
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-> 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_] := b[n, 0];
a /@ Range[0, 29] (* Jean-François Alcover, Mar 30 2021, 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 *)
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.
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