A227580 Number of lattice paths from {n}^3 to {0}^3 using steps that decrement one component such that for each point (p_1,p_2,p_3) we have p_1<=p_2<=p_3.

1, 1, 14, 290, 7680, 238636, 8285506, 312077474, 12509563082, 526701471002, 23076216957520, 1044813920439200, 48630132961189400, 2317337976558074760, 112689430179458971738, 5577655817793682738378, 280392321290875174774106, 14290804691034216155457274

Offset: 0

Author

Alois P. Heinz, Jul 16 2013

Keywords

• nonn

Examples

			a(2) = 14: [(2,2,2),(0,2,2),(0,0,2),(0,0,0)], [(2,2,2),(0,2,2),(0,0,2),(0,0,1),(0,0,0)], [(2,2,2),(0,2,2),(0,1,2),(0,0,2),(0,0,0)], [(2,2,2),(0,2,2),(0,1,2),(0,0,2),(0,0,1),(0,0,0)], [(2,2,2),(0,2,2),(0,1,2),(0,1,1),(0,0,1),(0,0,0)], [(2,2,2),(1,2,2),(0,2,2),(0,0,2),(0,0,0)], [(2,2,2),(1,2,2),(0,2,2),(0,0,2),(0,0,1),(0,0,0)], [(2,2,2),(1,2,2),(0,2,2),(0,1,2),(0,0,2),(0,0,0)], [(2,2,2),(1,2,2),(0,2,2),(0,1,2),(0,0,2),(0,0,1),(0,0,0)], [(2,2,2),(1,2,2),(0,2,2),(0,1,2),(0,1,1),(0,0,1),(0,0,0)], [(2,2,2),(1,2,2),(1,1,2),(0,1,2),(0,0,2),(0,0,0)], [(2,2,2),(1,2,2),(1,1,2),(0,1,2),(0,0,2),(0,0,1),(0,0,0)], [(2,2,2),(1,2,2),(1,1,2),(0,1,2),(0,1,1),(0,0,1),(0,0,0)], [(2,2,2),(1,2,2),(1,1,2),(1,1,1),(0,1,1),(0,0,1),(0,0,0)].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Column k=3 of A227578.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, [1, 1, 14][n+1],
      ((n+1)*(665*n^3-1433*n^2+980*n-204) *a(n-1)
       -(n-2)*(1615*n^3-3218*n^2+1521*n-342) *a(n-2)
       +192*(5*n-1)*(n-3)*(n-2)^2 *a(n-3)) /
       (2*(n+2)*(5*n-6)*(n+1)^2))
    end:
seq(a(n), n=0..30);

Mathematica

b[l_] := b[l] = If[l[[-1]] == 0, 1, Sum[Sum[b[ReplacePart[l, i -> j]], {j, If[i == 1, 0, l[[i-1]]], l[[i]] - 1}], {i, 1, Length[l]}]];
a[n_] := b[Array[n&, 3]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz in A227578 *)

Formula

a(n) ~ 2^(6*n+10)/(sqrt(3)*Pi*(5*n)^4). - Vaclav Kotesovec, Jul 18 2013