A333682 Number of nonnegative lattice paths from (0,0) to (4n+3,0) such that slopes of adjacent steps differ by one, assuming zero slope before and after the paths.

1, 3, 16, 119, 1070, 10751, 116287, 1326581, 15756587, 193181910, 2429921124, 31216684816, 408198225495, 5418728779290, 72871393962150, 991102308239835, 13613940451015378, 188650695857473559, 2634681336798911129, 37054660535787380825, 524449965598846642847

Offset: 0

Author

Alois P. Heinz, Apr 01 2020

Keywords

• nonn

Comments

The maximal height in all paths of length 4n+3 is (n+1)^2 = A000290(n+1).

The maximal area under all paths of length 4n+3 is 2*(n+1)^3 = A033431(n+1).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Alois P. Heinz, Animation of a(3) = 119 paths
• Alois P. Heinz, Plot of a(3) = 119 paths
• Alois P. Heinz, Plot of a(4) = 1070 paths
• Alois P. Heinz, Plot of a(5) = 10751 paths

Crossrefs

Cf. A000290, A004767, A033431, A333647.

Programs

Maple

b:= proc(x, y, t) option remember; `if`(x=0, 1, add(`if`(j=t, 0,
      b(x-1, y+j, j)), j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
    end:
a:= n-> b(4*n+3, 0$2):
seq(a(n), n=0..23);