A333678 Total area under all nonnegative lattice paths from (0,0) to (n,0) such that slopes of adjacent steps differ by at most one, assuming zero slope before and after the paths.

0, 0, 0, 2, 7, 22, 64, 196, 574, 1762, 5379, 16378, 49380, 148892, 449004, 1353718, 4076150, 12267160, 36903433, 110979048, 333628384, 1002722482, 3013085711, 9052404522, 27192329061, 81671691634, 245271884478, 736513920180, 2211445194899, 6639545054310

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..250

Alois P. Heinz, Animation of A333647(9) = 169 paths with total area a(9) = 1762

Wikipedia, Counting lattice paths

Maple

b:= proc(x, y, t) option remember; `if`(x=0, [1, 0],
      add((p-> p+[0, p[1]*(y+j/2)])(b(x-1, y+j, j)),
           j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
    end:
a:= n-> b(n, 0$2)[2]:
seq(a(n), n=0..38);

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[x == 0, {1, 0},
     Sum[Function[p, p + {0, If[p === 0, 0, p[[1]]]*(y + j/2)}][
     b[x-1, y+j, j]], {j, Max[t-1, -y], Min[x(x-1)/2-y, t+1]}]];
a[n_] := b[n, 0, 0][[2]];
a /@ Range[0, 38] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)

Crossrefs

Cf. A333647, A333679, A333680.

Sequence in context: A198888 A364539 A084264 * A088211 A071684 A290917

Adjacent sequences: A333675 A333676 A333677 * A333679 A333680 A333681

Keyword

nonn

Author

Alois P. Heinz, Apr 01 2020

Status

approved