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%I #12 Dec 20 2020 07:48:07
%S 1,1,14,290,7680,238636,8285506,312077474,12509563082,526701471002,
%T 23076216957520,1044813920439200,48630132961189400,
%U 2317337976558074760,112689430179458971738,5577655817793682738378,280392321290875174774106,14290804691034216155457274
%N 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.
%H Alois P. Heinz, <a href="/A227580/b227580.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) ~ 2^(6*n+10)/(sqrt(3)*Pi*(5*n)^4). - _Vaclav Kotesovec_, Jul 18 2013
%e 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)].
%p a:= proc(n) option remember; `if`(n<3, [1, 1, 14][n+1],
%p ((n+1)*(665*n^3-1433*n^2+980*n-204) *a(n-1)
%p -(n-2)*(1615*n^3-3218*n^2+1521*n-342) *a(n-2)
%p +192*(5*n-1)*(n-3)*(n-2)^2 *a(n-3)) /
%p (2*(n+2)*(5*n-6)*(n+1)^2))
%p end:
%p seq(a(n), n=0..30);
%t 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]}]];
%t a[n_] := b[Array[n&, 3]];
%t a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 20 2020, after _Alois P. Heinz_ in A227578 *)
%Y Column k=3 of A227578.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jul 16 2013
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