A333679 Sum of the heights of 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, 1, 3, 8, 20, 53, 137, 375, 1035, 2878, 7988, 22308, 62642, 176692, 499818, 1418228, 4035568, 11512449, 32916181, 94313011, 270757747, 778694171, 2243200705, 6471953522, 18699169766, 54098598824, 156706773404, 454457344755, 1319382151919, 3834346819731

Offset: 0

Author

Alois P. Heinz, Apr 01 2020

Keywords

• nonn

Comments

The maximal height in all paths of length n is floor(ceil(n/2)^2/4) = A008642(n-3) for n>2.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Alois P. Heinz, Animation of A333647(9) = 169 paths with height sum a(9) = 375
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A008642, A333647, A333678, A333680.

Programs

Maple

b:= proc(x, y, t, h) option remember;
      `if`(x=0, h, add(b(x-1, y+j, j, max(h, y)),
         j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..32);

Mathematica

b[x_, y_, t_, h_] := b[x, y, t, h] =
     If[x == 0, h, Sum[b[x - 1, y + j, j, Max[h, y]],
     {j, Max[t - 1, -y], Min[x(x - 1)/2 - y, t + 1]}]];
a[n_] := b[n, 0, 0, 0];
a /@ Range[0, 32] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)