A333498
Sum of the heights of all Motzkin paths of length n.
Original entry on oeis.org
0, 0, 1, 3, 9, 25, 70, 196, 552, 1560, 4423, 12573, 35826, 102310, 292786, 839554, 2411945, 6941593, 20011328, 57779038, 167069317, 483739961, 1402413161, 4070537585, 11827842021, 34403798725, 100167396088, 291903951462, 851380987390, 2485175809878
Offset: 0
-
b:= proc(x, y, h) option remember; `if`(x=0, h, add(
b(x-1, y+j, max(h, y)), j=-min(1, y)..min(1, x-y-1)))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..35);
-
b[x_, y_, h_] := b[x, y, h] = If[x == 0, h, Sum[b[x - 1, y + j, Max[h, y]], {j, -Min[1, y], Min[1, x - y - 1]}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 35] (* Jean-François Alcover, May 10 2020, after Maple *)
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 *)
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 *)
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 *)
Showing 1-5 of 5 results.