A225041
Number of lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps U=(1,1), D=(1,-1), H=(1,0) and S=(0,1).
Original entry on oeis.org
1, 1, 3, 9, 35, 145, 659, 3137, 15619, 80177, 422595, 2273633, 12447667, 69138193, 388784259, 2209440945, 12671782579, 73260414481, 426545078627, 2499059841249, 14723542302627, 87181150961361, 518554078448339, 3097007445391441, 18565515801339827
Offset: 0
a(0) = 1: the empty path.
a(1) = 1: H.
a(2) = 3: HH, UD, HSD.
a(3) = 9: HHH, UDH, HSDH, UHD, HSHD, HUD, HHSD, UDSD, HSDSD.
-
b:= proc(x, y) option remember; `if`(y>x, 0, `if`(x=0, 1,
b(x-1, y)+`if`(y>0, b(x-1, y-1)+b(x, y-1), 0)+b(x-1, y+1)))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);
-
b[x_, y_] := b[x, y] = If[y>x, 0, If[x==0, 1, b[x-1, y]+If[y>0, b[x-1, y-1] + b[x, y-1], 0] + b[x-1, y+1]]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 29 2017, translated from Maple *)
A224769
Number of lattice paths from (0,0) to (n,n) that do not go below the x-axis or above the diagonal x=y and consist of steps U=(1,1), D=(1,-1) and S=(0,1).
Original entry on oeis.org
1, 1, 2, 7, 33, 184, 1142, 7629, 53750, 394157, 2981546, 23117242, 182867360, 1470714606, 11993628444, 98967634147, 824958769631, 6937180941468, 58785077008641, 501520244718945, 4304433733010962, 37142428443486254, 322042675618484973, 2804409601249038670
Offset: 0
a(2) = 2: UDSS, UU.
a(3) = 7: UDSDSSS, UDUSS, UDSSDSS, UUDSS, UDSUS, UDSSU, UUU.
-
b:= proc(x, y) option remember; `if`(y>x, 0, `if`(x=0, 1,
`if`(y>0, b(x, y-1)+b(x-1, y-1), 0)+b(x-1, y+1)))
end:
a:= n-> b(n, n):
seq(a(n), n=0..30);
-
b[x_, y_] := b[x, y] = If[y > x, 0, If[x == 0, 1, If[y > 0, b[x, y - 1] + b[x - 1, y - 1], 0] + b[x - 1, y + 1]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)
A286427
Total number of nodes summed over all lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps U=(1,1), D=(1,-1) and S=(0,1).
Original entry on oeis.org
1, 0, 3, 5, 24, 78, 325, 1272, 5373, 22572, 97762, 425716, 1882062, 8375064, 37601643, 169773435, 771096972, 3518065767, 16119884297, 74125225732, 341970676466, 1582133854847, 7338641255894, 34117910537671, 158946919835352, 741884929970516, 3468677541274922
Offset: 0
-
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, [1$2],
(p-> p+[0, p[1]])(b(x, y-1)+b(x-1, y-1)+b(x-1, y+1))))
end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);
Showing 1-3 of 3 results.