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

Author

Alois P. Heinz, Apr 01 2020

Keywords

• nonn

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

Crossrefs

Cf. A333647, A333679, A333680.

Programs

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 *)