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

0,2

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

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);

Crossrefs

Cf. A000290, A004767, A033431, A333647.

Sequence in context: A302701 A190633 A220352 * A125807 A125222 A199670

Adjacent sequences: A333679 A333680 A333681 * A333683 A333684 A333685

Keyword

nonn

Author

Alois P. Heinz, Apr 01 2020

Status

approved